Properties

Label 4-608e2-1.1-c1e2-0-17
Degree $4$
Conductor $369664$
Sign $1$
Analytic cond. $23.5700$
Root an. cond. $2.20338$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 3·9-s − 4·11-s − 10·17-s − 2·19-s − 10·25-s + 14·27-s + 8·33-s − 16·41-s + 16·43-s − 5·49-s + 20·51-s + 4·57-s − 2·59-s − 26·67-s + 18·73-s + 20·75-s − 4·81-s − 20·83-s − 24·89-s + 28·97-s + 12·99-s − 30·107-s − 36·113-s − 10·121-s + 32·123-s + 127-s + ⋯
L(s)  = 1  − 1.15·3-s − 9-s − 1.20·11-s − 2.42·17-s − 0.458·19-s − 2·25-s + 2.69·27-s + 1.39·33-s − 2.49·41-s + 2.43·43-s − 5/7·49-s + 2.80·51-s + 0.529·57-s − 0.260·59-s − 3.17·67-s + 2.10·73-s + 2.30·75-s − 4/9·81-s − 2.19·83-s − 2.54·89-s + 2.84·97-s + 1.20·99-s − 2.90·107-s − 3.38·113-s − 0.909·121-s + 2.88·123-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 369664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(369664\)    =    \(2^{10} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(23.5700\)
Root analytic conductor: \(2.20338\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 369664,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
19$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
17$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
59$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
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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

−8.346693816906368374834545094569, −7.87598676263742878936605339825, −7.18297506166153508271072468025, −6.72377104864689556277323751664, −6.21150502900434643631796858377, −5.90719710332910393553929521875, −5.41676012615338774687665516924, −4.99527268882455447172353841146, −4.41313931480294198592546395456, −3.93346432196685798564362812764, −2.89182953598080292522833763992, −2.56005012019129152265351485884, −1.78091799751097958894751684261, 0, 0, 1.78091799751097958894751684261, 2.56005012019129152265351485884, 2.89182953598080292522833763992, 3.93346432196685798564362812764, 4.41313931480294198592546395456, 4.99527268882455447172353841146, 5.41676012615338774687665516924, 5.90719710332910393553929521875, 6.21150502900434643631796858377, 6.72377104864689556277323751664, 7.18297506166153508271072468025, 7.87598676263742878936605339825, 8.346693816906368374834545094569

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