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Properties

Label 2-936-104.99-c1-0-54
Degree $2$
Conductor $936$
Sign $-0.988 + 0.154i$
Analytic cond. $7.47399$
Root an. cond. $2.73386$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.849 − 1.13i)2-s + (−0.556 + 1.92i)4-s + (1.67 − 1.67i)5-s + (−1.16 − 1.16i)7-s + (2.64 − 1.00i)8-s + (−3.31 − 0.469i)10-s + (0.391 − 0.391i)11-s + (−3.59 + 0.311i)13-s + (−0.326 + 2.29i)14-s + (−3.38 − 2.13i)16-s − 2.95i·17-s + (−0.785 − 0.785i)19-s + (2.28 + 4.14i)20-s + (−0.776 − 0.110i)22-s − 2.14·23-s + ⋯
L(s)  = 1  + (−0.600 − 0.799i)2-s + (−0.278 + 0.960i)4-s + (0.747 − 0.747i)5-s + (−0.438 − 0.438i)7-s + (0.935 − 0.354i)8-s + (−1.04 − 0.148i)10-s + (0.118 − 0.118i)11-s + (−0.996 + 0.0863i)13-s + (−0.0871 + 0.613i)14-s + (−0.845 − 0.534i)16-s − 0.717i·17-s + (−0.180 − 0.180i)19-s + (0.510 + 0.926i)20-s + (−0.165 − 0.0234i)22-s − 0.447·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.154i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 + 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.988 + 0.154i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ -0.988 + 0.154i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0611716 - 0.788288i\)
\(L(\frac12)\) \(\approx\) \(0.0611716 - 0.788288i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.849 + 1.13i)T \)
3 \( 1 \)
13 \( 1 + (3.59 - 0.311i)T \)
good5 \( 1 + (-1.67 + 1.67i)T - 5iT^{2} \)
7 \( 1 + (1.16 + 1.16i)T + 7iT^{2} \)
11 \( 1 + (-0.391 + 0.391i)T - 11iT^{2} \)
17 \( 1 + 2.95iT - 17T^{2} \)
19 \( 1 + (0.785 + 0.785i)T + 19iT^{2} \)
23 \( 1 + 2.14T + 23T^{2} \)
29 \( 1 + 9.69iT - 29T^{2} \)
31 \( 1 + (-1.16 + 1.16i)T - 31iT^{2} \)
37 \( 1 + (-1.87 - 1.87i)T + 37iT^{2} \)
41 \( 1 + (-1.21 - 1.21i)T + 41iT^{2} \)
43 \( 1 - 6.64iT - 43T^{2} \)
47 \( 1 + (5.19 + 5.19i)T + 47iT^{2} \)
53 \( 1 + 13.9iT - 53T^{2} \)
59 \( 1 + (8.86 - 8.86i)T - 59iT^{2} \)
61 \( 1 + 5.77iT - 61T^{2} \)
67 \( 1 + (5.85 + 5.85i)T + 67iT^{2} \)
71 \( 1 + (5.33 - 5.33i)T - 71iT^{2} \)
73 \( 1 + (5.90 - 5.90i)T - 73iT^{2} \)
79 \( 1 - 1.82iT - 79T^{2} \)
83 \( 1 + (3.29 + 3.29i)T + 83iT^{2} \)
89 \( 1 + (-7.48 + 7.48i)T - 89iT^{2} \)
97 \( 1 + (-4.94 - 4.94i)T + 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.829319064578456244069460300038, −9.118181784535768001042711816693, −8.166486727978611979588331323483, −7.33577551562859603960303366270, −6.29742436209023209844886260172, −5.03722139233168232308326907796, −4.21119458304873733069832869674, −2.91906358256492819312902318066, −1.86179601577268498023651004531, −0.43760084899077024804606562655, 1.75486494588434133867953651966, 2.91240959273919121039263003979, 4.50679883758830005585365447864, 5.62784201069263683186813013691, 6.24863298895123308575868640497, 7.02290163166823499801849109895, 7.83210712521536358047205013348, 8.928385812014811109884749935036, 9.489733404670453309989102196871, 10.40373407156402576705807140843

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