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Properties

Label 2-930-155.123-c1-0-14
Degree $2$
Conductor $930$
Sign $0.818 + 0.574i$
Analytic cond. $7.42608$
Root an. cond. $2.72508$
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  + (−0.707 + 0.707i)2-s + (0.707 − 0.707i)3-s − 1.00i·4-s + (−2.23 − 0.0168i)5-s + 1.00i·6-s + (−2.83 + 2.83i)7-s + (0.707 + 0.707i)8-s − 1.00i·9-s + (1.59 − 1.56i)10-s + 1.81i·11-s + (−0.707 − 0.707i)12-s + (3.73 − 3.73i)13-s − 4.01i·14-s + (−1.59 + 1.56i)15-s − 1.00·16-s + (−0.414 − 0.414i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.408 − 0.408i)3-s − 0.500i·4-s + (−0.999 − 0.00754i)5-s + 0.408i·6-s + (−1.07 + 1.07i)7-s + (0.250 + 0.250i)8-s − 0.333i·9-s + (0.503 − 0.496i)10-s + 0.546i·11-s + (−0.204 − 0.204i)12-s + (1.03 − 1.03i)13-s − 1.07i·14-s + (−0.411 + 0.405i)15-s − 0.250·16-s + (−0.100 − 0.100i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.818 + 0.574i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.818 + 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.818 + 0.574i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.818 + 0.574i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.848543 - 0.268010i\)
\(L(\frac12)\) \(\approx\) \(0.848543 - 0.268010i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (2.23 + 0.0168i)T \)
31 \( 1 + (-0.282 + 5.56i)T \)
good7 \( 1 + (2.83 - 2.83i)T - 7iT^{2} \)
11 \( 1 - 1.81iT - 11T^{2} \)
13 \( 1 + (-3.73 + 3.73i)T - 13iT^{2} \)
17 \( 1 + (0.414 + 0.414i)T + 17iT^{2} \)
19 \( 1 + 5.16iT - 19T^{2} \)
23 \( 1 + (-1.45 + 1.45i)T - 23iT^{2} \)
29 \( 1 + 1.79T + 29T^{2} \)
37 \( 1 + (-7.51 - 7.51i)T + 37iT^{2} \)
41 \( 1 - 8.55T + 41T^{2} \)
43 \( 1 + (-1.19 + 1.19i)T - 43iT^{2} \)
47 \( 1 + (-5.64 + 5.64i)T - 47iT^{2} \)
53 \( 1 + (10.0 - 10.0i)T - 53iT^{2} \)
59 \( 1 + 11.6iT - 59T^{2} \)
61 \( 1 + 14.9iT - 61T^{2} \)
67 \( 1 + (-8.55 + 8.55i)T - 67iT^{2} \)
71 \( 1 + 8.79T + 71T^{2} \)
73 \( 1 + (-8.90 + 8.90i)T - 73iT^{2} \)
79 \( 1 + 1.10T + 79T^{2} \)
83 \( 1 + (4.92 - 4.92i)T - 83iT^{2} \)
89 \( 1 - 2.43T + 89T^{2} \)
97 \( 1 + (-2.65 + 2.65i)T - 97iT^{2} \)
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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

−9.580467884844702865595565888737, −9.099359288817664090261495831103, −8.227115519770840187544984458739, −7.62929600132891073363870133247, −6.61735750364854786336734568210, −6.00177850689372904477683130286, −4.75795851968812541679644917501, −3.41747960903142501051226554840, −2.52247309328304017447944057763, −0.58458283373137708381549776571, 1.09122678592054765387817549140, 2.94262063683427594789818504552, 3.91128112344054063312569678790, 4.11347154718789893608837881852, 5.98467440169559460551821583851, 7.02488212030316623924560838403, 7.73317036353246442663947676934, 8.653393257231171670900202760323, 9.294652366650686717813552227371, 10.18414904529099100047138685078

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