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Properties

Label 2-931-133.37-c0-0-0
Degree $2$
Conductor $931$
Sign $0.605 - 0.795i$
Analytic cond. $0.464629$
Root an. cond. $0.681637$
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  + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.499 + 0.866i)16-s + (1 + 1.73i)17-s + (0.5 − 0.866i)19-s + 0.999·20-s + (0.5 − 0.866i)23-s + 0.999·36-s − 43-s + (0.499 − 0.866i)44-s + (−0.499 − 0.866i)45-s + (−0.5 + 0.866i)47-s − 0.999·55-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.499 + 0.866i)16-s + (1 + 1.73i)17-s + (0.5 − 0.866i)19-s + 0.999·20-s + (0.5 − 0.866i)23-s + 0.999·36-s − 43-s + (0.499 − 0.866i)44-s + (−0.499 − 0.866i)45-s + (−0.5 + 0.866i)47-s − 0.999·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(931\)    =    \(7^{2} \cdot 19\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(0.464629\)
Root analytic conductor: \(0.681637\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{931} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 931,\ (\ :0),\ 0.605 - 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7816787239\)
\(L(\frac12)\) \(\approx\) \(0.7816787239\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
19 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (0.5 + 0.866i)T^{2} \)
3 \( 1 + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
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

−10.51965155021030884811589356441, −9.696833605043405474458211810637, −8.737757858825089411608211277743, −7.88583062254236102062964680503, −6.94914445839036822933054455115, −6.10500281671357042875532660945, −5.11880320498645872229286258147, −4.24823142207975782074527625819, −3.04881507901113434033220382517, −1.66431586233228758296887567353, 0.846219027675123186127661269350, 3.16011202294472921603183823811, 3.62655782214096131430972004391, 4.83871484909189759861215890156, 5.62429856210009566655797598174, 6.90565084269467785278640599694, 7.83315833468756567680668419051, 8.494899466022207883699125956180, 9.212757235108250744678764577279, 9.772764656172235479344526227283

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