Properties

Label 4-442e2-1.1-c1e2-0-10
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 4·9-s + 6·13-s + 16-s − 6·17-s + 4·19-s + 6·25-s − 4·36-s + 6·47-s + 6·49-s + 6·52-s − 18·53-s + 64-s + 16·67-s − 6·68-s + 4·76-s + 7·81-s + 12·83-s + 12·89-s + 6·100-s + 6·101-s + 18·103-s − 24·117-s − 6·121-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 1/2·4-s − 4/3·9-s + 1.66·13-s + 1/4·16-s − 1.45·17-s + 0.917·19-s + 6/5·25-s − 2/3·36-s + 0.875·47-s + 6/7·49-s + 0.832·52-s − 2.47·53-s + 1/8·64-s + 1.95·67-s − 0.727·68-s + 0.458·76-s + 7/9·81-s + 1.31·83-s + 1.27·89-s + 3/5·100-s + 0.597·101-s + 1.77·103-s − 2.21·117-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-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: \(0\)
Selberg data: \((4,\ 195364,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.792583395\)
\(L(\frac12)\) \(\approx\) \(1.792583395\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 6 T + p T^{2} \)
17$C_2$ \( 1 + 6 T + p T^{2} \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 66 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 36 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 120 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2^2$ \( 1 + 84 T^{2} + p^{2} T^{4} \)
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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

−9.001914773115974583203848522141, −8.736572086339827158671453510721, −8.141834107497585332472799022786, −7.84759208992761170875308390425, −7.04298287930287894214265601513, −6.66707570251208196621371810600, −6.12246861270718633440591600232, −5.86354748296126130781815579433, −5.12610515730511007367473727857, −4.67077919007395755241080931682, −3.77093975066548650218504104674, −3.31930059373198088739235828712, −2.69991880444065614847166302319, −1.96859010153033373289347055070, −0.873052954370341321930701237571, 0.873052954370341321930701237571, 1.96859010153033373289347055070, 2.69991880444065614847166302319, 3.31930059373198088739235828712, 3.77093975066548650218504104674, 4.67077919007395755241080931682, 5.12610515730511007367473727857, 5.86354748296126130781815579433, 6.12246861270718633440591600232, 6.66707570251208196621371810600, 7.04298287930287894214265601513, 7.84759208992761170875308390425, 8.141834107497585332472799022786, 8.736572086339827158671453510721, 9.001914773115974583203848522141

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