Properties

Label 4-98e2-1.1-c3e2-0-7
Degree $4$
Conductor $9604$
Sign $1$
Analytic cond. $33.4336$
Root an. cond. $2.40461$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 12·4-s + 32·8-s + 34·9-s + 40·11-s + 80·16-s + 136·18-s + 160·22-s + 96·23-s − 162·25-s − 332·29-s + 192·32-s + 408·36-s − 156·37-s + 872·43-s + 480·44-s + 384·46-s − 648·50-s + 124·53-s − 1.32e3·58-s + 448·64-s + 1.16e3·67-s − 1.08e3·71-s + 1.08e3·72-s − 624·74-s − 1.36e3·79-s + 427·81-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 1.41·8-s + 1.25·9-s + 1.09·11-s + 5/4·16-s + 1.78·18-s + 1.55·22-s + 0.870·23-s − 1.29·25-s − 2.12·29-s + 1.06·32-s + 17/9·36-s − 0.693·37-s + 3.09·43-s + 1.64·44-s + 1.23·46-s − 1.83·50-s + 0.321·53-s − 3.00·58-s + 7/8·64-s + 2.11·67-s − 1.81·71-s + 1.78·72-s − 0.980·74-s − 1.93·79-s + 0.585·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9604\)    =    \(2^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(33.4336\)
Root analytic conductor: \(2.40461\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{98} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9604,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(5.760469644\)
\(L(\frac12)\) \(\approx\) \(5.760469644\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{2} \)
7 \( 1 \)
good3$C_2^2$ \( 1 - 34 T^{2} + p^{6} T^{4} \)
5$C_2^2$ \( 1 + 162 T^{2} + p^{6} T^{4} \)
11$C_2$ \( ( 1 - 20 T + p^{3} T^{2} )^{2} \)
13$C_2^2$ \( 1 + 82 T^{2} + p^{6} T^{4} \)
17$C_2^2$ \( 1 + 6658 T^{2} + p^{6} T^{4} \)
19$C_2^2$ \( 1 + 13630 T^{2} + p^{6} T^{4} \)
23$C_2$ \( ( 1 - 48 T + p^{3} T^{2} )^{2} \)
29$C_2$ \( ( 1 + 166 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 16990 T^{2} + p^{6} T^{4} \)
37$C_2$ \( ( 1 + 78 T + p^{3} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 17390 T^{2} + p^{6} T^{4} \)
43$C_2$ \( ( 1 - 436 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 165054 T^{2} + p^{6} T^{4} \)
53$C_2$ \( ( 1 - 62 T + p^{3} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 32850 T^{2} + p^{6} T^{4} \)
61$C_2^2$ \( 1 + 379954 T^{2} + p^{6} T^{4} \)
67$C_2$ \( ( 1 - 580 T + p^{3} T^{2} )^{2} \)
71$C_2$ \( ( 1 + 544 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 417586 T^{2} + p^{6} T^{4} \)
79$C_2$ \( ( 1 + 680 T + p^{3} T^{2} )^{2} \)
83$C_2^2$ \( 1 + 1104766 T^{2} + p^{6} T^{4} \)
89$C_2^2$ \( 1 - 842862 T^{2} + p^{6} T^{4} \)
97$C_2^2$ \( 1 + 1394146 T^{2} + p^{6} T^{4} \)
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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

−13.59963797485483083534632429630, −13.18031487263087406742530126025, −12.65761964241543160645639262883, −12.39851490830612553633858556115, −11.60381728625889836671756696456, −11.29996879555470691184140180085, −10.69905435151612401309135613329, −10.00587691118305216349327883130, −9.418810470744951450183158449036, −8.894540399069061869708057750005, −7.79143107836201692503870437619, −7.22634835454322682372038762076, −6.94647134629169526302865347732, −5.96000826408606816820597607154, −5.62322633178832773910526720905, −4.59366733869488539456384880253, −4.03583657467447267655429694013, −3.56321171856643082977611247433, −2.26949176586125277843156415495, −1.33862522665358230379204874497, 1.33862522665358230379204874497, 2.26949176586125277843156415495, 3.56321171856643082977611247433, 4.03583657467447267655429694013, 4.59366733869488539456384880253, 5.62322633178832773910526720905, 5.96000826408606816820597607154, 6.94647134629169526302865347732, 7.22634835454322682372038762076, 7.79143107836201692503870437619, 8.894540399069061869708057750005, 9.418810470744951450183158449036, 10.00587691118305216349327883130, 10.69905435151612401309135613329, 11.29996879555470691184140180085, 11.60381728625889836671756696456, 12.39851490830612553633858556115, 12.65761964241543160645639262883, 13.18031487263087406742530126025, 13.59963797485483083534632429630

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