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Properties

Label 2-936-24.11-c1-0-35
Degree $2$
Conductor $936$
Sign $0.576 + 0.816i$
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  + (1.40 + 0.160i)2-s + (1.94 + 0.450i)4-s − 3.43·5-s − 1.48i·7-s + (2.66 + 0.944i)8-s + (−4.82 − 0.550i)10-s − 4.11i·11-s − i·13-s + (0.237 − 2.08i)14-s + (3.59 + 1.75i)16-s − 6.57i·17-s + 7.48·19-s + (−6.69 − 1.54i)20-s + (0.659 − 5.77i)22-s + 2.95·23-s + ⋯
L(s)  = 1  + (0.993 + 0.113i)2-s + (0.974 + 0.225i)4-s − 1.53·5-s − 0.560i·7-s + (0.942 + 0.334i)8-s + (−1.52 − 0.174i)10-s − 1.24i·11-s − 0.277i·13-s + (0.0635 − 0.557i)14-s + (0.898 + 0.438i)16-s − 1.59i·17-s + 1.71·19-s + (−1.49 − 0.345i)20-s + (0.140 − 1.23i)22-s + 0.616·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.576 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.576 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.576 + 0.816i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (755, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ 0.576 + 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.99409 - 1.03319i\)
\(L(\frac12)\) \(\approx\) \(1.99409 - 1.03319i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 - 0.160i)T \)
3 \( 1 \)
13 \( 1 + iT \)
good5 \( 1 + 3.43T + 5T^{2} \)
7 \( 1 + 1.48iT - 7T^{2} \)
11 \( 1 + 4.11iT - 11T^{2} \)
17 \( 1 + 6.57iT - 17T^{2} \)
19 \( 1 - 7.48T + 19T^{2} \)
23 \( 1 - 2.95T + 23T^{2} \)
29 \( 1 + 2.46T + 29T^{2} \)
31 \( 1 + 4.42iT - 31T^{2} \)
37 \( 1 - 9.06iT - 37T^{2} \)
41 \( 1 + 2.36iT - 41T^{2} \)
43 \( 1 + 11.0T + 43T^{2} \)
47 \( 1 - 4.15T + 47T^{2} \)
53 \( 1 + 11.6T + 53T^{2} \)
59 \( 1 - 9.65iT - 59T^{2} \)
61 \( 1 + 13.0iT - 61T^{2} \)
67 \( 1 + 5.33T + 67T^{2} \)
71 \( 1 + 4.14T + 71T^{2} \)
73 \( 1 - 2.91T + 73T^{2} \)
79 \( 1 + 6.01iT - 79T^{2} \)
83 \( 1 + 7.31iT - 83T^{2} \)
89 \( 1 - 1.95iT - 89T^{2} \)
97 \( 1 - 15.3T + 97T^{2} \)
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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

−10.18621393009550784802693841364, −8.934361507603669374269762391130, −7.74041153139324484188007040401, −7.54830517097745200017925398112, −6.54795928687248229280794701723, −5.32495795591441452554232627736, −4.61297785097270773418355319503, −3.41905480107356144445930200130, −3.11551833925162751828089252506, −0.812796941234807894541506578991, 1.66690324679854164776666455906, 3.11614367172072569826210212541, 3.92416642281879964161977466701, 4.74577248737522699965919846517, 5.65188245270175997246189021824, 6.88779260005402057503617164461, 7.45423176646329434667152480333, 8.264887946144987334091610595067, 9.418447088252725189807778161527, 10.46860978525980359980388190484

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