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Properties

Label 4-395136-1.1-c1e2-0-11
Degree $4$
Conductor $395136$
Sign $1$
Analytic cond. $25.1942$
Root an. cond. $2.24039$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s + 9-s + 8·11-s + 2·25-s − 12·29-s + 4·37-s + 49-s − 4·53-s − 63-s + 24·67-s + 8·71-s − 8·77-s + 16·79-s + 81-s + 8·99-s + 4·109-s + 4·113-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 0.377·7-s + 1/3·9-s + 2.41·11-s + 2/5·25-s − 2.22·29-s + 0.657·37-s + 1/7·49-s − 0.549·53-s − 0.125·63-s + 2.93·67-s + 0.949·71-s − 0.911·77-s + 1.80·79-s + 1/9·81-s + 0.804·99-s + 0.383·109-s + 0.376·113-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 395136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 395136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(395136\)    =    \(2^{7} \cdot 3^{2} \cdot 7^{3}\)
Sign: $1$
Analytic conductor: \(25.1942\)
Root analytic conductor: \(2.24039\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 395136,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.120295274\)
\(L(\frac12)\) \(\approx\) \(2.120295274\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$ \( 1 + T \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.11.ai_bm
13$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.13.a_o
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.19.a_ak
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.a_as
29$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.m_da
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.a_ac
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.ae_ck
41$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.41.a_aby
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.a_cs
47$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.47.a_abi
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.53.e_bu
59$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.59.a_bm
61$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.61.a_abi
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.67.ay_ks
71$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.71.ai_fm
73$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.73.a_o
79$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + p T^{2} ) \) 2.79.aq_gc
83$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.83.a_acw
89$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.89.a_be
97$C_2^2$ \( 1 + 174 T^{2} + p^{2} T^{4} \) 2.97.a_gs
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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

−8.794982359450167923789887590868, −8.204420559086206556134899851533, −7.73694030079543785567613120202, −7.12734401822485341175405975378, −6.81924134749275848987947664797, −6.36402506409128597376286431421, −6.01435317827498382717859862828, −5.33115285448310005780944318016, −4.81054544024340501455446265148, −4.02305041623616919160605193005, −3.80328204016342582460563312505, −3.36433690062115085321106139907, −2.32327110558899929025387741249, −1.68354813351536804100064609537, −0.864085438211432364167736092630, 0.864085438211432364167736092630, 1.68354813351536804100064609537, 2.32327110558899929025387741249, 3.36433690062115085321106139907, 3.80328204016342582460563312505, 4.02305041623616919160605193005, 4.81054544024340501455446265148, 5.33115285448310005780944318016, 6.01435317827498382717859862828, 6.36402506409128597376286431421, 6.81924134749275848987947664797, 7.12734401822485341175405975378, 7.73694030079543785567613120202, 8.204420559086206556134899851533, 8.794982359450167923789887590868

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