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Properties

Label 2-936-936.779-c1-0-42
Degree $2$
Conductor $936$
Sign $0.975 - 0.218i$
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 + (0.486 + 1.66i)3-s + (0.999 − 1.73i)4-s + (−3.85 − 2.22i)5-s + (−1.77 − 1.69i)6-s + (0.257 + 0.446i)7-s + 2.82i·8-s + (−2.52 + 1.61i)9-s + 6.29·10-s + (3.36 + 0.820i)12-s + (−1.80 + 3.12i)13-s + (−0.630 − 0.364i)14-s + (1.82 − 7.49i)15-s + (−2.00 − 3.46i)16-s − 4.35i·17-s + (1.95 − 3.76i)18-s + ⋯
L(s)  = 1  + (−0.866 + 0.499i)2-s + (0.280 + 0.959i)3-s + (0.499 − 0.866i)4-s + (−1.72 − 0.996i)5-s + (−0.722 − 0.690i)6-s + (0.0973 + 0.168i)7-s + 0.999i·8-s + (−0.842 + 0.538i)9-s + 1.99·10-s + (0.971 + 0.236i)12-s + (−0.499 + 0.866i)13-s + (−0.168 − 0.0973i)14-s + (0.471 − 1.93i)15-s + (−0.500 − 0.866i)16-s − 1.05i·17-s + (0.460 − 0.887i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.218i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.975 - 0.218i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (779, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ 0.975 - 0.218i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.631921 + 0.0697804i\)
\(L(\frac12)\) \(\approx\) \(0.631921 + 0.0697804i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.486 - 1.66i)T \)
13 \( 1 + (1.80 - 3.12i)T \)
good5 \( 1 + (3.85 + 2.22i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.257 - 0.446i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 4.35iT - 17T^{2} \)
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 \( 1 - 9.80T + 37T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.41 - 2.44i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-10.1 + 5.85i)T + (23.5 - 40.7i)T^{2} \)
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 \( 1 - 8.27iT - 71T^{2} \)
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

−9.679405471536809751504091962884, −9.223747927899119900184281453197, −8.430811278891412240579651797799, −7.83573977398577099670352708985, −7.08107103833247748205898095451, −5.61170789122894094610907191692, −4.67663451995353990321245145925, −4.10837188079460222897697104516, −2.58976974780529855098227710646, −0.55435861138399705999303412668, 0.865028516905345819325377929889, 2.59416208395479618831446094486, 3.26426280845272104025974565385, 4.24835210798857072631635422868, 6.22712179214990064264982103604, 7.03949681689927821973993964532, 7.74613751645411937440884356160, 8.078092925998988549933232663994, 8.936666883766593383299094412717, 10.32798888747937275928459923047

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