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Properties

Label 2-936-104.101-c1-0-31
Degree $2$
Conductor $936$
Sign $-0.0308 - 0.999i$
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  + (0.114 + 1.40i)2-s + (−1.97 + 0.323i)4-s + 3.80·5-s + (1.38 + 0.798i)7-s + (−0.683 − 2.74i)8-s + (0.437 + 5.36i)10-s + (−0.318 − 0.550i)11-s + (−0.717 + 3.53i)13-s + (−0.966 + 2.04i)14-s + (3.79 − 1.27i)16-s + (−0.777 + 1.34i)17-s + (3.07 − 5.31i)19-s + (−7.50 + 1.23i)20-s + (0.739 − 0.511i)22-s + (2.48 + 4.30i)23-s + ⋯
L(s)  = 1  + (0.0812 + 0.996i)2-s + (−0.986 + 0.161i)4-s + 1.70·5-s + (0.522 + 0.301i)7-s + (−0.241 − 0.970i)8-s + (0.138 + 1.69i)10-s + (−0.0958 − 0.166i)11-s + (−0.199 + 0.979i)13-s + (−0.258 + 0.545i)14-s + (0.947 − 0.319i)16-s + (−0.188 + 0.326i)17-s + (0.704 − 1.21i)19-s + (−1.67 + 0.275i)20-s + (0.157 − 0.109i)22-s + (0.517 + 0.896i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0308 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0308 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.0308 - 0.999i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ -0.0308 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46136 + 1.50720i\)
\(L(\frac12)\) \(\approx\) \(1.46136 + 1.50720i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.114 - 1.40i)T \)
3 \( 1 \)
13 \( 1 + (0.717 - 3.53i)T \)
good5 \( 1 - 3.80T + 5T^{2} \)
7 \( 1 + (-1.38 - 0.798i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.318 + 0.550i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.777 - 1.34i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.07 + 5.31i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.48 - 4.30i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.48 - 1.43i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.35iT - 31T^{2} \)
37 \( 1 + (-5.36 - 9.28i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.325 - 0.188i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.61 - 2.66i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 10.6iT - 47T^{2} \)
53 \( 1 - 12.4iT - 53T^{2} \)
59 \( 1 + (4.06 - 7.03i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.32 + 3.07i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.10 + 1.91i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (8.08 + 4.66i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 2.57iT - 73T^{2} \)
79 \( 1 - 10.4T + 79T^{2} \)
83 \( 1 + 13.6T + 83T^{2} \)
89 \( 1 + (11.1 - 6.42i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.0755 + 0.0436i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.844455762949004649886083460082, −9.322636778989297271620418887376, −8.770524432897363664721121835172, −7.59642681880109892920035290870, −6.73650960613567926184834375466, −5.95193538619083879750861893491, −5.25854153273280046321829620818, −4.45589492516748907947741936160, −2.81947770356916099115595107000, −1.49495325270094095980422978036, 1.15073267515141203366227660628, 2.18494767428429779144653077966, 3.13494993770885295166576539665, 4.54882520545550884455810363889, 5.41227060922740314846039017425, 6.01438859041168954745761402006, 7.43164322954780861547648081426, 8.440809191116285018368789026192, 9.371773851232504589098798800984, 9.933927149034015513165655902857

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