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Properties

Label 2-936-104.83-c1-0-66
Degree $2$
Conductor $936$
Sign $-0.224 - 0.974i$
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  + (0.140 − 1.40i)2-s + (−1.96 − 0.394i)4-s + (−3.08 − 3.08i)5-s + (2.85 − 2.85i)7-s + (−0.829 + 2.70i)8-s + (−4.77 + 3.91i)10-s + (−2.57 − 2.57i)11-s + (−3.51 + 0.821i)13-s + (−3.62 − 4.42i)14-s + (3.68 + 1.54i)16-s − 1.07i·17-s + (1.46 − 1.46i)19-s + (4.83 + 7.27i)20-s + (−3.98 + 3.26i)22-s + 5.52·23-s + ⋯
L(s)  = 1  + (0.0990 − 0.995i)2-s + (−0.980 − 0.197i)4-s + (−1.38 − 1.38i)5-s + (1.08 − 1.08i)7-s + (−0.293 + 0.956i)8-s + (−1.51 + 1.23i)10-s + (−0.776 − 0.776i)11-s + (−0.973 + 0.227i)13-s + (−0.967 − 1.18i)14-s + (0.922 + 0.386i)16-s − 0.259i·17-s + (0.335 − 0.335i)19-s + (1.08 + 1.62i)20-s + (−0.850 + 0.696i)22-s + 1.15·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.224 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.224 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.224 - 0.974i$
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: \((2,\ 936,\ (\ :1/2),\ -0.224 - 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.413019 + 0.518820i\)
\(L(\frac12)\) \(\approx\) \(0.413019 + 0.518820i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.140 + 1.40i)T \)
3 \( 1 \)
13 \( 1 + (3.51 - 0.821i)T \)
good5 \( 1 + (3.08 + 3.08i)T + 5iT^{2} \)
7 \( 1 + (-2.85 + 2.85i)T - 7iT^{2} \)
11 \( 1 + (2.57 + 2.57i)T + 11iT^{2} \)
17 \( 1 + 1.07iT - 17T^{2} \)
19 \( 1 + (-1.46 + 1.46i)T - 19iT^{2} \)
23 \( 1 - 5.52T + 23T^{2} \)
29 \( 1 + 0.512iT - 29T^{2} \)
31 \( 1 + (2.85 + 2.85i)T + 31iT^{2} \)
37 \( 1 + (-1.97 + 1.97i)T - 37iT^{2} \)
41 \( 1 + (-0.0336 + 0.0336i)T - 41iT^{2} \)
43 \( 1 - 3.49iT - 43T^{2} \)
47 \( 1 + (6.45 - 6.45i)T - 47iT^{2} \)
53 \( 1 - 8.04iT - 53T^{2} \)
59 \( 1 + (1.23 + 1.23i)T + 59iT^{2} \)
61 \( 1 - 4.66iT - 61T^{2} \)
67 \( 1 + (-2.03 + 2.03i)T - 67iT^{2} \)
71 \( 1 + (-5.60 - 5.60i)T + 71iT^{2} \)
73 \( 1 + (5.73 + 5.73i)T + 73iT^{2} \)
79 \( 1 + 6.65iT - 79T^{2} \)
83 \( 1 + (0.153 - 0.153i)T - 83iT^{2} \)
89 \( 1 + (11.3 + 11.3i)T + 89iT^{2} \)
97 \( 1 + (-8.80 + 8.80i)T - 97iT^{2} \)
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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

−9.408342010643168296308137635615, −8.696337827876677692503577700713, −7.82551006083746157773344684895, −7.48003263167321194240814777681, −5.34688734612222073289646948179, −4.72704967731462697076728973868, −4.19509910233748768808525091353, −3.04010441712524204719865046588, −1.30386841458030310588889949464, −0.33646941125536493560013249829, 2.48139959033077415430041595091, 3.54044140503209918819693373727, 4.76546671213374985725976484787, 5.32438871260952648666954656657, 6.68023216298334678701055393981, 7.34356437899266070659707503375, 7.944544286144651228510095953957, 8.551781028104944628218968778697, 9.781708837062942274383663613871, 10.63947951294266730578684638279

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