Properties

Label 4-442e2-1.1-c1e2-0-44
Degree $4$
Conductor $195364$
Sign $-1$
Analytic cond. $12.4565$
Root an. cond. $1.87866$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 2·9-s − 2·13-s + 16-s − 6·17-s − 2·25-s − 2·36-s + 8·43-s − 14·49-s − 2·52-s − 12·53-s + 64-s − 6·68-s − 5·81-s − 2·100-s − 12·101-s + 16·103-s + 4·117-s − 14·121-s + 127-s + 131-s + 137-s + 139-s − 2·144-s + 149-s + 151-s + 12·153-s + ⋯
L(s)  = 1  + 1/2·4-s − 2/3·9-s − 0.554·13-s + 1/4·16-s − 1.45·17-s − 2/5·25-s − 1/3·36-s + 1.21·43-s − 2·49-s − 0.277·52-s − 1.64·53-s + 1/8·64-s − 0.727·68-s − 5/9·81-s − 1/5·100-s − 1.19·101-s + 1.57·103-s + 0.369·117-s − 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/6·144-s + 0.0819·149-s + 0.0813·151-s + 0.970·153-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195364 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195364 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(195364\)    =    \(2^{2} \cdot 13^{2} \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(12.4565\)
Root analytic conductor: \(1.87866\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{195364} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 195364,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
13$C_2$ \( 1 + 2 T + p T^{2} \)
17$C_2$ \( 1 + 6 T + p T^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 130 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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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

−8.853790750332823305364314039604, −8.429777542683242916058960709044, −7.78940998615953799581372571836, −7.58270190082111774390777293052, −6.82790761321645748134751661393, −6.44160438391538980810023068904, −6.07367302421691823683409874535, −5.38109929106858249234873578043, −4.85509792889292551724272820250, −4.31973642956798773198823316705, −3.61000021734553610589767848756, −2.85586016126681762407877228226, −2.38168843253748205404354975710, −1.56726240961227233040592054552, 0, 1.56726240961227233040592054552, 2.38168843253748205404354975710, 2.85586016126681762407877228226, 3.61000021734553610589767848756, 4.31973642956798773198823316705, 4.85509792889292551724272820250, 5.38109929106858249234873578043, 6.07367302421691823683409874535, 6.44160438391538980810023068904, 6.82790761321645748134751661393, 7.58270190082111774390777293052, 7.78940998615953799581372571836, 8.429777542683242916058960709044, 8.853790750332823305364314039604

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