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Properties

Label 2-936-24.11-c1-0-3
Degree $2$
Conductor $936$
Sign $-0.793 + 0.608i$
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  + (−0.141 + 1.40i)2-s + (−1.95 − 0.398i)4-s − 0.802·5-s + 1.93i·7-s + (0.838 − 2.70i)8-s + (0.113 − 1.12i)10-s + 1.56i·11-s + i·13-s + (−2.72 − 0.274i)14-s + (3.68 + 1.56i)16-s + 3.54i·17-s + 4.68·19-s + (1.57 + 0.320i)20-s + (−2.19 − 0.221i)22-s − 8.86·23-s + ⋯
L(s)  = 1  + (−0.100 + 0.994i)2-s + (−0.979 − 0.199i)4-s − 0.359·5-s + 0.732i·7-s + (0.296 − 0.955i)8-s + (0.0359 − 0.357i)10-s + 0.471i·11-s + 0.277i·13-s + (−0.728 − 0.0733i)14-s + (0.920 + 0.390i)16-s + 0.860i·17-s + 1.07·19-s + (0.351 + 0.0715i)20-s + (−0.468 − 0.0471i)22-s − 1.84·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.793 + 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.793 + 0.608i$
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: \((2,\ 936,\ (\ :1/2),\ -0.793 + 0.608i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.160063 - 0.471682i\)
\(L(\frac12)\) \(\approx\) \(0.160063 - 0.471682i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.141 - 1.40i)T \)
3 \( 1 \)
13 \( 1 - iT \)
good5 \( 1 + 0.802T + 5T^{2} \)
7 \( 1 - 1.93iT - 7T^{2} \)
11 \( 1 - 1.56iT - 11T^{2} \)
17 \( 1 - 3.54iT - 17T^{2} \)
19 \( 1 - 4.68T + 19T^{2} \)
23 \( 1 + 8.86T + 23T^{2} \)
29 \( 1 + 8.14T + 29T^{2} \)
31 \( 1 + 10.3iT - 31T^{2} \)
37 \( 1 - 7.05iT - 37T^{2} \)
41 \( 1 + 4.31iT - 41T^{2} \)
43 \( 1 + 1.21T + 43T^{2} \)
47 \( 1 + 10.6T + 47T^{2} \)
53 \( 1 - 1.19T + 53T^{2} \)
59 \( 1 - 8.98iT - 59T^{2} \)
61 \( 1 - 2.66iT - 61T^{2} \)
67 \( 1 + 9.46T + 67T^{2} \)
71 \( 1 + 2.27T + 71T^{2} \)
73 \( 1 + 12.1T + 73T^{2} \)
79 \( 1 + 0.0205iT - 79T^{2} \)
83 \( 1 - 0.956iT - 83T^{2} \)
89 \( 1 - 4.61iT - 89T^{2} \)
97 \( 1 - 15.1T + 97T^{2} \)
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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

−10.15821893009386931514806709859, −9.632755836315550724148696721376, −8.757776390774516307301862009840, −7.87880812909737255518302145332, −7.39474135372321856900639495097, −6.09272537030242564210014219368, −5.69831324872078819207029754999, −4.45128041767609954945252920457, −3.65291402261512869646364042814, −1.89751692366739635724140189940, 0.24204424657584819474284460033, 1.70437118671024131510236380582, 3.18516040849055519527179443342, 3.84580060595569812684037690502, 4.91665986514782733589699023196, 5.87755632471993273930884855305, 7.35866407667391371752339149555, 7.893655938273865468624112265251, 8.901206497140850638489225665992, 9.776164893709763227961014920147

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