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Properties

Label 2-930-155.92-c1-0-6
Degree $2$
Conductor $930$
Sign $0.421 - 0.907i$
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 + (−1.64 − 1.50i)5-s − 1.00i·6-s + (2.19 + 2.19i)7-s + (0.707 − 0.707i)8-s + 1.00i·9-s + (0.0993 + 2.23i)10-s + 2.28i·11-s + (−0.707 + 0.707i)12-s + (−0.645 − 0.645i)13-s − 3.10i·14-s + (−0.0993 − 2.23i)15-s − 1.00·16-s + (−1.02 + 1.02i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.408 + 0.408i)3-s + 0.500i·4-s + (−0.737 − 0.675i)5-s − 0.408i·6-s + (0.830 + 0.830i)7-s + (0.250 − 0.250i)8-s + 0.333i·9-s + (0.0314 + 0.706i)10-s + 0.689i·11-s + (−0.204 + 0.204i)12-s + (−0.179 − 0.179i)13-s − 0.830i·14-s + (−0.0256 − 0.576i)15-s − 0.250·16-s + (−0.249 + 0.249i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.421 - 0.907i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.421 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.421 - 0.907i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.421 - 0.907i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.915561 + 0.584371i\)
\(L(\frac12)\) \(\approx\) \(0.915561 + 0.584371i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.64 + 1.50i)T \)
31 \( 1 + (-5.39 - 1.36i)T \)
good7 \( 1 + (-2.19 - 2.19i)T + 7iT^{2} \)
11 \( 1 - 2.28iT - 11T^{2} \)
13 \( 1 + (0.645 + 0.645i)T + 13iT^{2} \)
17 \( 1 + (1.02 - 1.02i)T - 17iT^{2} \)
19 \( 1 - 2.57iT - 19T^{2} \)
23 \( 1 + (0.646 + 0.646i)T + 23iT^{2} \)
29 \( 1 + 6.94T + 29T^{2} \)
37 \( 1 + (5.23 - 5.23i)T - 37iT^{2} \)
41 \( 1 - 4.11T + 41T^{2} \)
43 \( 1 + (-7.37 - 7.37i)T + 43iT^{2} \)
47 \( 1 + (2.53 + 2.53i)T + 47iT^{2} \)
53 \( 1 + (-8.61 - 8.61i)T + 53iT^{2} \)
59 \( 1 - 1.51iT - 59T^{2} \)
61 \( 1 - 7.53iT - 61T^{2} \)
67 \( 1 + (0.158 + 0.158i)T + 67iT^{2} \)
71 \( 1 - 3.65T + 71T^{2} \)
73 \( 1 + (-1.99 - 1.99i)T + 73iT^{2} \)
79 \( 1 + 12.7T + 79T^{2} \)
83 \( 1 + (-7.17 - 7.17i)T + 83iT^{2} \)
89 \( 1 + 10.1T + 89T^{2} \)
97 \( 1 + (4.14 + 4.14i)T + 97iT^{2} \)
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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

−10.12263337133622647400407022007, −9.292773552224769273491802459785, −8.579984956252693663998454314614, −8.035994030386385727581677722375, −7.24066624354976490845831975676, −5.65775800441483020392914278633, −4.68807580580473162199016576349, −3.95628888697997632561783466461, −2.64970059565476722451491829705, −1.51402350151939471810813842579, 0.59977564617364934747376630214, 2.19538631411182010495630952911, 3.56333924161150695278819222237, 4.51803033800707790263479735068, 5.78733335429422606200603891971, 6.94603591486323592134321042616, 7.36296564773109796375274092312, 8.111977319610523132472945986208, 8.819064881388230197002159775230, 9.836760063548350386331819811177

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