Properties

Label 4-98e2-1.1-c9e2-0-2
Degree $4$
Conductor $9604$
Sign $1$
Analytic cond. $2547.57$
Root an. cond. $7.10447$
Motivic weight $9$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 16·2-s − 156·3-s + 870·5-s + 2.49e3·6-s + 4.09e3·8-s + 1.96e4·9-s − 1.39e4·10-s + 5.61e4·11-s − 3.56e5·13-s − 1.35e5·15-s − 6.55e4·16-s − 2.47e5·17-s − 3.14e5·18-s + 3.15e5·19-s − 8.98e5·22-s − 2.04e5·23-s − 6.38e5·24-s + 1.95e6·25-s + 5.69e6·26-s − 5.41e6·27-s − 7.68e6·29-s + 2.17e6·30-s − 1.30e6·31-s − 8.75e6·33-s + 3.96e6·34-s − 4.30e6·37-s − 5.04e6·38-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.11·3-s + 0.622·5-s + 0.786·6-s + 0.353·8-s + 9-s − 0.440·10-s + 1.15·11-s − 3.45·13-s − 0.692·15-s − 1/4·16-s − 0.719·17-s − 0.707·18-s + 0.555·19-s − 0.817·22-s − 0.152·23-s − 0.393·24-s + 25-s + 2.44·26-s − 1.96·27-s − 2.01·29-s + 0.489·30-s − 0.254·31-s − 1.28·33-s + 0.508·34-s − 0.377·37-s − 0.392·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9604\)    =    \(2^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(2547.57\)
Root analytic conductor: \(7.10447\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9604,\ (\ :9/2, 9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(0.4192202377\)
\(L(\frac12)\) \(\approx\) \(0.4192202377\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 + p^{4} T + p^{8} T^{2} \)
7 \( 1 \)
good3$C_2^2$ \( 1 + 52 p T + 517 p^{2} T^{2} + 52 p^{10} T^{3} + p^{18} T^{4} \)
5$C_2^2$ \( 1 - 174 p T - 47849 p^{2} T^{2} - 174 p^{10} T^{3} + p^{18} T^{4} \)
11$C_2^2$ \( 1 - 56148 T + 794650213 T^{2} - 56148 p^{9} T^{3} + p^{18} T^{4} \)
13$C_2$ \( ( 1 + 178094 T + p^{9} T^{2} )^{2} \)
17$C_2^2$ \( 1 + 247662 T - 57251410253 T^{2} + 247662 p^{9} T^{3} + p^{18} T^{4} \)
19$C_2^2$ \( 1 - 315380 T - 223223153379 T^{2} - 315380 p^{9} T^{3} + p^{18} T^{4} \)
23$C_2^2$ \( 1 + 204504 T - 1759330775447 T^{2} + 204504 p^{9} T^{3} + p^{18} T^{4} \)
29$C_2$ \( ( 1 + 3840450 T + p^{9} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 1309408 T - 24725072850207 T^{2} + 1309408 p^{9} T^{3} + p^{18} T^{4} \)
37$C_2^2$ \( 1 + 4307078 T - 111410818896993 T^{2} + 4307078 p^{9} T^{3} + p^{18} T^{4} \)
41$C_2$ \( ( 1 + 1512042 T + p^{9} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 33670604 T + p^{9} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 10581072 T - 1007171388433583 T^{2} + 10581072 p^{9} T^{3} + p^{18} T^{4} \)
53$C_2^2$ \( 1 + 16616214 T - 3023665024108337 T^{2} + 16616214 p^{9} T^{3} + p^{18} T^{4} \)
59$C_2^2$ \( 1 - 112235100 T + 3933721853355061 T^{2} - 112235100 p^{9} T^{3} + p^{18} T^{4} \)
61$C_2^2$ \( 1 + 33197218 T - 10592090809894617 T^{2} + 33197218 p^{9} T^{3} + p^{18} T^{4} \)
67$C_2^2$ \( 1 - 121372252 T - 12475310840743443 T^{2} - 121372252 p^{9} T^{3} + p^{18} T^{4} \)
71$C_2$ \( ( 1 + 387172728 T + p^{9} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 255240074 T + 6275908667257563 T^{2} - 255240074 p^{9} T^{3} + p^{18} T^{4} \)
79$C_2^2$ \( 1 + 492101840 T + 122312624948767281 T^{2} + 492101840 p^{9} T^{3} + p^{18} T^{4} \)
83$C_2$ \( ( 1 - 457420236 T + p^{9} T^{2} )^{2} \)
89$C_2^2$ \( 1 + 31809510 T - 349344558781045109 T^{2} + 31809510 p^{9} T^{3} + p^{18} T^{4} \)
97$C_2$ \( ( 1 - 673532062 T + p^{9} T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.42618533208250034157618683046, −11.56152427274424393701526045891, −11.51084106646393716660099209668, −10.68876986535942152673602801881, −9.989842143996338542206901112982, −9.814354446817957578618307436175, −9.138671112114830826373639797673, −9.026426954429088764799998942723, −7.51621485258398969615633629428, −7.39433316765406525786434332505, −7.04258101790204369601194387944, −6.00301921097847079995136776963, −5.64185849062370363961445982942, −4.72214063866599154512290574051, −4.59536392295298442343675299839, −3.50579211832962700167123225234, −2.13324057302124454803657399365, −2.09864247647028490021212066738, −0.940247271813379156911731598131, −0.25316047158794046944426652672, 0.25316047158794046944426652672, 0.940247271813379156911731598131, 2.09864247647028490021212066738, 2.13324057302124454803657399365, 3.50579211832962700167123225234, 4.59536392295298442343675299839, 4.72214063866599154512290574051, 5.64185849062370363961445982942, 6.00301921097847079995136776963, 7.04258101790204369601194387944, 7.39433316765406525786434332505, 7.51621485258398969615633629428, 9.026426954429088764799998942723, 9.138671112114830826373639797673, 9.814354446817957578618307436175, 9.989842143996338542206901112982, 10.68876986535942152673602801881, 11.51084106646393716660099209668, 11.56152427274424393701526045891, 12.42618533208250034157618683046

Graph of the $Z$-function along the critical line