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Properties

Label 2-936-104.51-c0-0-3
Degree $2$
Conductor $936$
Sign $0.707 + 0.707i$
Analytic cond. $0.467124$
Root an. cond. $0.683465$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)2-s + 1.00i·4-s + 1.41·5-s + (0.707 − 0.707i)8-s + (−1.00 − 1.00i)10-s − 1.41i·11-s + i·13-s − 1.00·16-s + 1.41i·20-s + (−1.00 + 1.00i)22-s + 1.00·25-s + (0.707 − 0.707i)26-s + (0.707 + 0.707i)32-s + (1.00 − 1.00i)40-s + 1.41i·41-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + 1.00i·4-s + 1.41·5-s + (0.707 − 0.707i)8-s + (−1.00 − 1.00i)10-s − 1.41i·11-s + i·13-s − 1.00·16-s + 1.41i·20-s + (−1.00 + 1.00i)22-s + 1.00·25-s + (0.707 − 0.707i)26-s + (0.707 + 0.707i)32-s + (1.00 − 1.00i)40-s + 1.41i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(0.467124\)
Root analytic conductor: \(0.683465\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :0),\ 0.707 + 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8985599368\)
\(L(\frac12)\) \(\approx\) \(0.8985599368\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 \)
13 \( 1 - iT \)
good5 \( 1 - 1.41T + T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + 1.41iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - 1.41iT - T^{2} \)
43 \( 1 - 2T + T^{2} \)
47 \( 1 + 1.41T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + 1.41iT - T^{2} \)
61 \( 1 + 2iT - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + 1.41T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - 2iT - T^{2} \)
83 \( 1 - 1.41iT - T^{2} \)
89 \( 1 + 1.41iT - T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.941344671948642931395341374649, −9.492632470578869766926768017058, −8.740276548204964376387323216734, −7.940610028328123241764287139327, −6.66266794673490988752760789457, −6.03875703056062715577473795242, −4.81457792766906044631072160940, −3.50308456887480262453885378270, −2.45821725759523605317365796984, −1.39311631696663406780618495937, 1.54766558668733351766556994757, 2.56020530367154507535736744319, 4.51024711297295743116149855049, 5.47008027145559052696867685239, 6.05774544553801474052447238272, 7.06487917414651145903737168298, 7.73155105728424811410318266311, 8.886466944763009545648017162188, 9.477135905428355058476686180342, 10.26629355822319063502023309289

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