Properties

Label 4-442e2-1.1-c1e2-0-35
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  − 2·2-s + 3·4-s − 4·8-s + 2·9-s + 5·16-s − 4·18-s − 2·19-s − 6·25-s − 6·32-s + 6·36-s + 4·38-s − 18·43-s − 6·47-s + 6·49-s + 12·50-s − 12·59-s + 7·64-s + 16·67-s − 8·72-s − 6·76-s − 5·81-s + 6·83-s + 36·86-s − 12·89-s + 12·94-s − 12·98-s − 18·100-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 1.41·8-s + 2/3·9-s + 5/4·16-s − 0.942·18-s − 0.458·19-s − 6/5·25-s − 1.06·32-s + 36-s + 0.648·38-s − 2.74·43-s − 0.875·47-s + 6/7·49-s + 1.69·50-s − 1.56·59-s + 7/8·64-s + 1.95·67-s − 0.942·72-s − 0.688·76-s − 5/9·81-s + 0.658·83-s + 3.88·86-s − 1.27·89-s + 1.23·94-s − 1.21·98-s − 9/5·100-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$C_1$ \( ( 1 + T )^{2} \)
13$C_2$ \( 1 + p T^{2} \)
17$C_2$ \( 1 + p T^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
73$C_2^2$ \( 1 - 120 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
97$C_2^2$ \( 1 + 48 T^{2} + p^{2} T^{4} \)
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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

−8.724351216062232006519368047670, −8.501485855778936029164314069587, −7.967650561792832175821844926670, −7.61103762683312191658192406853, −7.01973641268209185204175508441, −6.59580079404043476915505263193, −6.23156379551221269269417631691, −5.47831814327383916749991575110, −4.95446343002521254928591052575, −4.12911958370613051159502903670, −3.56928520905311824141162256533, −2.78129939460403044898592782356, −1.95956316416164389803489247357, −1.38857387629553184445666491741, 0, 1.38857387629553184445666491741, 1.95956316416164389803489247357, 2.78129939460403044898592782356, 3.56928520905311824141162256533, 4.12911958370613051159502903670, 4.95446343002521254928591052575, 5.47831814327383916749991575110, 6.23156379551221269269417631691, 6.59580079404043476915505263193, 7.01973641268209185204175508441, 7.61103762683312191658192406853, 7.967650561792832175821844926670, 8.501485855778936029164314069587, 8.724351216062232006519368047670

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