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Properties

Label 4-7098e2-1.1-c1e2-0-3
Degree $4$
Conductor $50381604$
Sign $1$
Analytic cond. $3212.37$
Root an. cond. $7.52846$
Motivic weight $1$
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  − 2·2-s + 2·3-s + 3·4-s − 5-s − 4·6-s − 2·7-s − 4·8-s + 3·9-s + 2·10-s + 8·11-s + 6·12-s + 4·14-s − 2·15-s + 5·16-s + 6·17-s − 6·18-s + 2·19-s − 3·20-s − 4·21-s − 16·22-s − 3·23-s − 8·24-s + 5·25-s + 4·27-s − 6·28-s + 9·29-s + 4·30-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.447·5-s − 1.63·6-s − 0.755·7-s − 1.41·8-s + 9-s + 0.632·10-s + 2.41·11-s + 1.73·12-s + 1.06·14-s − 0.516·15-s + 5/4·16-s + 1.45·17-s − 1.41·18-s + 0.458·19-s − 0.670·20-s − 0.872·21-s − 3.41·22-s − 0.625·23-s − 1.63·24-s + 25-s + 0.769·27-s − 1.13·28-s + 1.67·29-s + 0.730·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50381604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50381604 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(50381604\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(3212.37\)
Root analytic conductor: \(7.52846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 50381604,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.724842720\)
\(L(\frac12)\) \(\approx\) \(2.724842720\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2$C_1$ \( ( 1 + T )^{2} \)
3$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
13 \( 1 \)
good5$C_2^2$ \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) 2.5.b_ae
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.11.ai_bm
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.17.ag_br
19$D_{4}$ \( 1 - 2 T - 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.19.ac_as
23$D_{4}$ \( 1 + 3 T + 34 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.23.d_bi
29$D_{4}$ \( 1 - 9 T + 64 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.29.aj_cm
31$D_{4}$ \( 1 + 5 T + 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.31.f_cc
37$D_{4}$ \( 1 - 7 T + 72 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.37.ah_cu
41$D_{4}$ \( 1 + 9 T + 88 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.41.j_dk
43$C_2^2$ \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) 2.43.b_abq
47$D_{4}$ \( 1 + 2 T + 38 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.47.c_bm
53$D_{4}$ \( 1 - 8 T + 65 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.53.ai_cn
59$D_{4}$ \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) 2.59.ab_ak
61$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.61.s_hv
67$D_{4}$ \( 1 - 13 T + 162 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.67.an_gg
71$D_{4}$ \( 1 + 3 T + 130 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.71.d_fa
73$D_{4}$ \( 1 + 19 T + 222 T^{2} + 19 p T^{3} + p^{2} T^{4} \) 2.73.t_io
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.79.q_io
83$D_{4}$ \( 1 - T + 38 T^{2} - p T^{3} + p^{2} T^{4} \) 2.83.ab_bm
89$D_{4}$ \( 1 + 9 T + 184 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.89.j_hc
97$D_{4}$ \( 1 + 14 T + 186 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.97.o_he
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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

−8.076855775688247442228217440280, −7.942793483557331915388758580715, −7.41989287424777244833473036236, −7.24902609699782249377691633453, −6.70442981158701344110969234067, −6.70032907335442660600515889677, −6.28759163635866726182130806331, −5.81438657650852241095066224780, −5.47556775664803999647498203844, −4.82803234800180857750011396590, −4.22972319632168507443973899713, −4.08028836443506289658098082777, −3.58953855335710328535501331403, −3.23303425284918098769209398022, −2.85183341623058143068866976509, −2.68038826018460051733773395230, −1.63708231583989981428054504986, −1.57543823826712693795743274494, −1.09358988712774501287963897065, −0.52807551743414018328827708778, 0.52807551743414018328827708778, 1.09358988712774501287963897065, 1.57543823826712693795743274494, 1.63708231583989981428054504986, 2.68038826018460051733773395230, 2.85183341623058143068866976509, 3.23303425284918098769209398022, 3.58953855335710328535501331403, 4.08028836443506289658098082777, 4.22972319632168507443973899713, 4.82803234800180857750011396590, 5.47556775664803999647498203844, 5.81438657650852241095066224780, 6.28759163635866726182130806331, 6.70032907335442660600515889677, 6.70442981158701344110969234067, 7.24902609699782249377691633453, 7.41989287424777244833473036236, 7.942793483557331915388758580715, 8.076855775688247442228217440280

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