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Properties

Label 2-936-104.101-c1-0-36
Degree $2$
Conductor $936$
Sign $0.635 + 0.772i$
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  + (−0.402 + 1.35i)2-s + (−1.67 − 1.09i)4-s − 0.607·5-s + (−1.20 − 0.695i)7-s + (2.15 − 1.83i)8-s + (0.244 − 0.823i)10-s + (1.88 + 3.26i)11-s + (−3.60 − 0.0168i)13-s + (1.42 − 1.35i)14-s + (1.62 + 3.65i)16-s + (−0.221 + 0.383i)17-s + (0.639 − 1.10i)19-s + (1.01 + 0.662i)20-s + (−5.18 + 1.24i)22-s + (−3.77 − 6.53i)23-s + ⋯
L(s)  = 1  + (−0.284 + 0.958i)2-s + (−0.838 − 0.545i)4-s − 0.271·5-s + (−0.455 − 0.262i)7-s + (0.761 − 0.648i)8-s + (0.0772 − 0.260i)10-s + (0.568 + 0.984i)11-s + (−0.999 − 0.00467i)13-s + (0.381 − 0.361i)14-s + (0.405 + 0.914i)16-s + (−0.0537 + 0.0931i)17-s + (0.146 − 0.254i)19-s + (0.227 + 0.148i)20-s + (−1.10 + 0.265i)22-s + (−0.787 − 1.36i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 + 0.772i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.635 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.635 + 0.772i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ 0.635 + 0.772i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.555242 - 0.262260i\)
\(L(\frac12)\) \(\approx\) \(0.555242 - 0.262260i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.402 - 1.35i)T \)
3 \( 1 \)
13 \( 1 + (3.60 + 0.0168i)T \)
good5 \( 1 + 0.607T + 5T^{2} \)
7 \( 1 + (1.20 + 0.695i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.88 - 3.26i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.221 - 0.383i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.639 + 1.10i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.77 + 6.53i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-8.53 + 4.92i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.17iT - 31T^{2} \)
37 \( 1 + (2.09 + 3.62i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-8.71 + 5.03i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.12 + 3.53i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 0.744iT - 47T^{2} \)
53 \( 1 + 1.24iT - 53T^{2} \)
59 \( 1 + (1.28 - 2.22i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.48 - 4.90i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.06 - 8.77i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.40 + 4.27i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 14.6iT - 73T^{2} \)
79 \( 1 + 12.7T + 79T^{2} \)
83 \( 1 + 6.19T + 83T^{2} \)
89 \( 1 + (0.530 - 0.306i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.74 + 3.31i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.973403253341850908739556387424, −9.063247669228827341002148619493, −8.116299555621289251953675119811, −7.34552645216964940988204340906, −6.67676085578809701561797156192, −5.81502146058354447848038206943, −4.57230143031821509450097047372, −4.04873145038466600239628967677, −2.25897619650909870782515363646, −0.33822897292366550356914908859, 1.36131273841544568479193527838, 2.84103296981791806465178791335, 3.56637364376851940137949427644, 4.68229686864523786025565095756, 5.71846090353748857002076922402, 6.88568515753550145912377251429, 7.956166718680951790568708984425, 8.614373522276165824327597637583, 9.590938542263223730509693999282, 10.01660279089200111246766273084

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