Properties

Label 4-608e2-1.1-c1e2-0-10
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 $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 3·9-s + 12·11-s + 6·17-s − 2·19-s − 10·25-s + 14·27-s − 24·33-s − 16·43-s − 13·49-s − 12·51-s + 4·57-s − 18·59-s − 10·67-s − 14·73-s + 20·75-s − 4·81-s + 12·83-s − 24·89-s − 20·97-s − 36·99-s + 18·107-s + 12·113-s + 86·121-s + 127-s + 32·129-s + 131-s + ⋯
L(s)  = 1  − 1.15·3-s − 9-s + 3.61·11-s + 1.45·17-s − 0.458·19-s − 2·25-s + 2.69·27-s − 4.17·33-s − 2.43·43-s − 1.85·49-s − 1.68·51-s + 0.529·57-s − 2.34·59-s − 1.22·67-s − 1.63·73-s + 2.30·75-s − 4/9·81-s + 1.31·83-s − 2.54·89-s − 2.03·97-s − 3.61·99-s + 1.74·107-s + 1.12·113-s + 7.81·121-s + 0.0887·127-s + 2.81·129-s + 0.0873·131-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: $\chi_{369664} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
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$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
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$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−8.666881218945426773456780708719, −8.059349923675553885391410616419, −7.49822591496797438401690336538, −6.79687210361750407728091466084, −6.44167245371229223240488859667, −6.04366496258479301724306565185, −5.93366234401884381484558908154, −5.20686401818851827647471141246, −4.58652942277425117027930400256, −4.07138629192131545636061302538, −3.41440775482142096356524512987, −3.12424893607844884324162519175, −1.68171654176783246558951167350, −1.34050914666900867420115229318, 0, 1.34050914666900867420115229318, 1.68171654176783246558951167350, 3.12424893607844884324162519175, 3.41440775482142096356524512987, 4.07138629192131545636061302538, 4.58652942277425117027930400256, 5.20686401818851827647471141246, 5.93366234401884381484558908154, 6.04366496258479301724306565185, 6.44167245371229223240488859667, 6.79687210361750407728091466084, 7.49822591496797438401690336538, 8.059349923675553885391410616419, 8.666881218945426773456780708719

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