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Properties

Label 2-936-104.99-c1-0-21
Degree $2$
Conductor $936$
Sign $0.660 - 0.750i$
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  + (−1.41 − 0.0632i)2-s + (1.99 + 0.178i)4-s + (1.61 − 1.61i)5-s + (2.08 + 2.08i)7-s + (−2.80 − 0.378i)8-s + (−2.39 + 2.18i)10-s + (0.0673 − 0.0673i)11-s + (−1.04 + 3.45i)13-s + (−2.80 − 3.07i)14-s + (3.93 + 0.712i)16-s + 7.18i·17-s + (2.30 + 2.30i)19-s + (3.51 − 2.93i)20-s + (−0.0994 + 0.0909i)22-s − 2.00·23-s + ⋯
L(s)  = 1  + (−0.998 − 0.0447i)2-s + (0.995 + 0.0894i)4-s + (0.724 − 0.724i)5-s + (0.786 + 0.786i)7-s + (−0.990 − 0.133i)8-s + (−0.756 + 0.691i)10-s + (0.0203 − 0.0203i)11-s + (−0.288 + 0.957i)13-s + (−0.750 − 0.821i)14-s + (0.984 + 0.178i)16-s + 1.74i·17-s + (0.528 + 0.528i)19-s + (0.786 − 0.656i)20-s + (−0.0212 + 0.0193i)22-s − 0.417·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.660 - 0.750i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.660 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.660 - 0.750i$
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: \((2,\ 936,\ (\ :1/2),\ 0.660 - 0.750i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07478 + 0.486102i\)
\(L(\frac12)\) \(\approx\) \(1.07478 + 0.486102i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0632i)T \)
3 \( 1 \)
13 \( 1 + (1.04 - 3.45i)T \)
good5 \( 1 + (-1.61 + 1.61i)T - 5iT^{2} \)
7 \( 1 + (-2.08 - 2.08i)T + 7iT^{2} \)
11 \( 1 + (-0.0673 + 0.0673i)T - 11iT^{2} \)
17 \( 1 - 7.18iT - 17T^{2} \)
19 \( 1 + (-2.30 - 2.30i)T + 19iT^{2} \)
23 \( 1 + 2.00T + 23T^{2} \)
29 \( 1 - 4.37iT - 29T^{2} \)
31 \( 1 + (2.08 - 2.08i)T - 31iT^{2} \)
37 \( 1 + (6.43 + 6.43i)T + 37iT^{2} \)
41 \( 1 + (-7.94 - 7.94i)T + 41iT^{2} \)
43 \( 1 + 10.4iT - 43T^{2} \)
47 \( 1 + (5.31 + 5.31i)T + 47iT^{2} \)
53 \( 1 - 1.41iT - 53T^{2} \)
59 \( 1 + (-3.96 + 3.96i)T - 59iT^{2} \)
61 \( 1 - 1.94iT - 61T^{2} \)
67 \( 1 + (-8.16 - 8.16i)T + 67iT^{2} \)
71 \( 1 + (0.00829 - 0.00829i)T - 71iT^{2} \)
73 \( 1 + (-0.419 + 0.419i)T - 73iT^{2} \)
79 \( 1 + 8.46iT - 79T^{2} \)
83 \( 1 + (-3.35 - 3.35i)T + 83iT^{2} \)
89 \( 1 + (1.90 - 1.90i)T - 89iT^{2} \)
97 \( 1 + (-8.76 - 8.76i)T + 97iT^{2} \)
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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

−10.04193375526750129755893216253, −9.146184631041036683705962952262, −8.695991147579761929255150095816, −7.955688450271726348488325076642, −6.87334120964194560101715554745, −5.85224156345010166568557426303, −5.22232641139158101519696063161, −3.75399092275884630780920436261, −2.06772097862696173513130048249, −1.55766362958475946865618777412, 0.789736566050338846045977921221, 2.25657289363042910020804558374, 3.13720060985524577363837541574, 4.80122911176575119697221561308, 5.80964487141304296048283919229, 6.81843413117890521084059133437, 7.50762993625013671834664348212, 8.114369989218533132158402782914, 9.379670741049812992746044168338, 9.869732368117910635016667040691

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