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Properties

Label 2-936-104.83-c1-0-45
Degree $2$
Conductor $936$
Sign $-0.987 + 0.154i$
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  + (−0.456 − 1.33i)2-s + (−1.58 + 1.22i)4-s + (1.04 + 1.04i)5-s + (−2.37 + 2.37i)7-s + (2.35 + 1.55i)8-s + (0.919 − 1.87i)10-s + (−2.03 − 2.03i)11-s + (−3.25 + 1.55i)13-s + (4.26 + 2.09i)14-s + (1.00 − 3.87i)16-s − 4.25i·17-s + (0.214 − 0.214i)19-s + (−2.92 − 0.375i)20-s + (−1.79 + 3.65i)22-s − 0.169·23-s + ⋯
L(s)  = 1  + (−0.323 − 0.946i)2-s + (−0.791 + 0.611i)4-s + (0.466 + 0.466i)5-s + (−0.896 + 0.896i)7-s + (0.834 + 0.551i)8-s + (0.290 − 0.592i)10-s + (−0.614 − 0.614i)11-s + (−0.902 + 0.430i)13-s + (1.13 + 0.559i)14-s + (0.252 − 0.967i)16-s − 1.03i·17-s + (0.0492 − 0.0492i)19-s + (−0.654 − 0.0838i)20-s + (−0.382 + 0.779i)22-s − 0.0353·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.154i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 + 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.987 + 0.154i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (811, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ -0.987 + 0.154i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0312047 - 0.400818i\)
\(L(\frac12)\) \(\approx\) \(0.0312047 - 0.400818i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.456 + 1.33i)T \)
3 \( 1 \)
13 \( 1 + (3.25 - 1.55i)T \)
good5 \( 1 + (-1.04 - 1.04i)T + 5iT^{2} \)
7 \( 1 + (2.37 - 2.37i)T - 7iT^{2} \)
11 \( 1 + (2.03 + 2.03i)T + 11iT^{2} \)
17 \( 1 + 4.25iT - 17T^{2} \)
19 \( 1 + (-0.214 + 0.214i)T - 19iT^{2} \)
23 \( 1 + 0.169T + 23T^{2} \)
29 \( 1 + 9.09iT - 29T^{2} \)
31 \( 1 + (4.96 + 4.96i)T + 31iT^{2} \)
37 \( 1 + (-2.37 + 2.37i)T - 37iT^{2} \)
41 \( 1 + (-3.95 + 3.95i)T - 41iT^{2} \)
43 \( 1 + 2.48iT - 43T^{2} \)
47 \( 1 + (0.869 - 0.869i)T - 47iT^{2} \)
53 \( 1 + 1.81iT - 53T^{2} \)
59 \( 1 + (3.97 + 3.97i)T + 59iT^{2} \)
61 \( 1 - 0.851iT - 61T^{2} \)
67 \( 1 + (8.69 - 8.69i)T - 67iT^{2} \)
71 \( 1 + (3.22 + 3.22i)T + 71iT^{2} \)
73 \( 1 + (-3.95 - 3.95i)T + 73iT^{2} \)
79 \( 1 - 10.5iT - 79T^{2} \)
83 \( 1 + (2.18 - 2.18i)T - 83iT^{2} \)
89 \( 1 + (-7.97 - 7.97i)T + 89iT^{2} \)
97 \( 1 + (10.2 - 10.2i)T - 97iT^{2} \)
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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

−9.537650635246349733224075903212, −9.325376640769531621180058133393, −8.159184863054242427439931590561, −7.25908093453984987176125319112, −6.09848546876104272733348745953, −5.26166314902779162687960441757, −4.01265715067070537114155115210, −2.66925389394637687074408774157, −2.42157349867579630985778902734, −0.20849072084785083827258483495, 1.47601583110510017153119738554, 3.31409554464282417961418117279, 4.56500408472438896530272069870, 5.32449292304486377192694570766, 6.27449459878823703357040839470, 7.18369375587576786405648829179, 7.71190401792412384841916446666, 8.837614745165396231962208904328, 9.548343619692452217873242325017, 10.26629328565791610616688658998

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