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Properties

Label 2-936-936.155-c1-0-87
Degree $2$
Conductor $936$
Sign $-0.298 - 0.954i$
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  + (1.22 + 0.707i)2-s + (1.19 + 1.25i)3-s + (0.999 + 1.73i)4-s + (−1.62 + 0.940i)5-s + (0.580 + 2.37i)6-s + (2.40 − 4.17i)7-s + 2.82i·8-s + (−0.136 + 2.99i)9-s − 2.66·10-s + (−0.972 + 3.32i)12-s + (1.80 + 3.12i)13-s + (5.90 − 3.40i)14-s + (−3.12 − 0.914i)15-s + (−2.00 + 3.46i)16-s + 3.88i·17-s + (−2.28 + 3.57i)18-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)2-s + (0.690 + 0.722i)3-s + (0.499 + 0.866i)4-s + (−0.728 + 0.420i)5-s + (0.236 + 0.971i)6-s + (0.910 − 1.57i)7-s + 0.999i·8-s + (−0.0454 + 0.998i)9-s − 0.841·10-s + (−0.280 + 0.959i)12-s + (0.499 + 0.866i)13-s + (1.57 − 0.910i)14-s + (−0.807 − 0.236i)15-s + (−0.500 + 0.866i)16-s + 0.943i·17-s + (−0.538 + 0.842i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.298 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.298 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.298 - 0.954i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ -0.298 - 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87714 + 2.55528i\)
\(L(\frac12)\) \(\approx\) \(1.87714 + 2.55528i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 - 0.707i)T \)
3 \( 1 + (-1.19 - 1.25i)T \)
13 \( 1 + (-1.80 - 3.12i)T \)
good5 \( 1 + (1.62 - 0.940i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2.40 + 4.17i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 3.88iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.47 + 9.48i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 11.1T + 37T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.84 + 8.38i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.284 + 0.164i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 16.8iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (48.5 + 84.0i)T^{2} \)
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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

−10.66200866709804789388728518606, −9.387066539152833996689907618171, −8.271878112244417673818085658987, −7.69269807757840756457994267612, −7.16196115421868909990344797423, −5.93005682694989197549830560659, −4.60887694493833819444797545534, −4.03031492117981564667384286792, −3.58598463271465342126576146223, −2.00430458522543591222576047926, 1.15833965292290226762107495671, 2.40862300958801647398926444610, 3.16488949385009448658934041938, 4.42289487209081941819314959372, 5.37160866920509072038548346104, 6.13426511369985487335844917485, 7.36112116133645424061051862106, 8.180602195611823436488949467335, 8.847382940881030046461439297472, 9.697322973249639752326743714415

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