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Properties

Label 2-936-104.77-c1-0-30
Degree $2$
Conductor $936$
Sign $-0.741 + 0.670i$
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  + (−0.273 − 1.38i)2-s + (−1.85 + 0.758i)4-s − 3.11·5-s + 2.77i·7-s + (1.55 + 2.36i)8-s + (0.850 + 4.32i)10-s + 2.56·11-s + (−0.546 − 3.56i)13-s + (3.85 − 0.758i)14-s + (2.85 − 2.80i)16-s + 5.70·17-s − 4.75·19-s + (5.76 − 2.36i)20-s + (−0.701 − 3.56i)22-s − 4·23-s + ⋯
L(s)  = 1  + (−0.193 − 0.981i)2-s + (−0.925 + 0.379i)4-s − 1.39·5-s + 1.04i·7-s + (0.550 + 0.834i)8-s + (0.269 + 1.36i)10-s + 0.774·11-s + (−0.151 − 0.988i)13-s + (1.02 − 0.202i)14-s + (0.712 − 0.701i)16-s + 1.38·17-s − 1.09·19-s + (1.28 − 0.527i)20-s + (−0.149 − 0.759i)22-s − 0.834·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.741 + 0.670i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.741 + 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.741 + 0.670i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ -0.741 + 0.670i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.239403 - 0.621658i\)
\(L(\frac12)\) \(\approx\) \(0.239403 - 0.621658i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.273 + 1.38i)T \)
3 \( 1 \)
13 \( 1 + (0.546 + 3.56i)T \)
good5 \( 1 + 3.11T + 5T^{2} \)
7 \( 1 - 2.77iT - 7T^{2} \)
11 \( 1 - 2.56T + 11T^{2} \)
17 \( 1 - 5.70T + 17T^{2} \)
19 \( 1 + 4.75T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 7.12iT - 29T^{2} \)
31 \( 1 + 3.60iT - 31T^{2} \)
37 \( 1 + 4.20T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 1.51iT - 43T^{2} \)
47 \( 1 + 2.77iT - 47T^{2} \)
53 \( 1 + 6.06iT - 53T^{2} \)
59 \( 1 + 4.75T + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 12.0T + 67T^{2} \)
71 \( 1 + 6.66iT - 71T^{2} \)
73 \( 1 + 14.9iT - 73T^{2} \)
79 \( 1 - 14.8T + 79T^{2} \)
83 \( 1 - 9.89T + 83T^{2} \)
89 \( 1 + 14.9iT - 89T^{2} \)
97 \( 1 + 3.89iT - 97T^{2} \)
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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

−9.828286022731661430505098586402, −8.924278315432554705469308491742, −8.085704517585026672570514388744, −7.74116913737963098694068099816, −6.16651137125773218247911642314, −5.12512998291423123967512743926, −4.00936451388314980510403079837, −3.36284269845356095765153520589, −2.14316793178640026629968038327, −0.40352731510118359990102467301, 1.17422864888259244450570961872, 3.71790040254393809981153890261, 4.05395563758347315538369780597, 5.07073893596819638393421426811, 6.45817572124121569657824147702, 7.05783270045936627562740314036, 7.76788133312851967361070269468, 8.463166734873376167469528034001, 9.347150627721566921691558834758, 10.30876051308000861547115368500

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