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Properties

Label 2-936-936.155-c1-0-45
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

\(L(1)\) \(\approx\) \(0.629701 + 0.857191i\)
\(L(\frac12)\) \(\approx\) \(0.629701 + 0.857191i\)
\(L(\frac{3}{2})\) 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

−10.08273093781587034637602529250, −9.503691653869516726569121729647, −8.624691266553716139331173664364, −8.512103046127972116266530887626, −7.11251804800898494551627533510, −5.91313961044408509725733067888, −5.12932645468869144710517407106, −3.55915763677239759804620720313, −2.77345095754245289149661901655, −1.88438883244663532908971572569, 0.57372674453422111982588229137, 1.97062315410781246674601753788, 3.04322427556740031401577847350, 4.43143749405385126532861667275, 6.06442152772336766219483026583, 6.68957442376076028056160601497, 7.24292972421398406181642910863, 7.914911706250605769971579792029, 9.141303513668190671813449842017, 9.710839718923083157758743039311

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