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Properties

Label 2-936-104.77-c1-0-21
Degree $2$
Conductor $936$
Sign $-0.196 - 0.980i$
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 + i)2-s − 2i·4-s + 5-s + 3i·7-s + (2 + 2i)8-s + (−1 + i)10-s + 2·11-s + (3 + 2i)13-s + (−3 − 3i)14-s − 4·16-s − 3·17-s − 2i·20-s + (−2 + 2i)22-s + 6·23-s − 4·25-s + (−5 + i)26-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s − i·4-s + 0.447·5-s + 1.13i·7-s + (0.707 + 0.707i)8-s + (−0.316 + 0.316i)10-s + 0.603·11-s + (0.832 + 0.554i)13-s + (−0.801 − 0.801i)14-s − 16-s − 0.727·17-s − 0.447i·20-s + (−0.426 + 0.426i)22-s + 1.25·23-s − 0.800·25-s + (−0.980 + 0.196i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.196 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.196 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.196 - 0.980i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ -0.196 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.747922 + 0.912318i\)
\(L(\frac12)\) \(\approx\) \(0.747922 + 0.912318i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
3 \( 1 \)
13 \( 1 + (-3 - 2i)T \)
good5 \( 1 - T + 5T^{2} \)
7 \( 1 - 3iT - 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 3T + 37T^{2} \)
41 \( 1 - 10iT - 41T^{2} \)
43 \( 1 - 9iT - 43T^{2} \)
47 \( 1 - 7iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 10T + 59T^{2} \)
61 \( 1 + 10iT - 61T^{2} \)
67 \( 1 - 12T + 67T^{2} \)
71 \( 1 - 5iT - 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 16T + 83T^{2} \)
89 \( 1 + 4iT - 89T^{2} \)
97 \( 1 - 18iT - 97T^{2} \)
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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

−9.896031036831754310614677895701, −9.318769480271139168092393886138, −8.720224063261848675880045306886, −7.953424199830714189751546535025, −6.69002984096417571674870515759, −6.21658840685772715266108046972, −5.37050853715856357629698359101, −4.28217468648720941394710763754, −2.56668023260271273998744267207, −1.41932102265810360053244028427, 0.77671772229382756802961712828, 1.95342585048035458596849589691, 3.41304881685524990367262677816, 4.07455294406761503892822765197, 5.39988461629087636259830217889, 6.82376974626189334107760792119, 7.21038028289056137379673801635, 8.489778513412188484963770881819, 8.963966574754652868723293127536, 9.992638986484286305177440986783

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