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Properties

Label 2-930-155.92-c1-0-24
Degree $2$
Conductor $930$
Sign $0.846 + 0.532i$
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 − 0.0964i)5-s − 1.00i·6-s + (−1.13 − 1.13i)7-s + (−0.707 + 0.707i)8-s + 1.00i·9-s + (1.64 + 1.51i)10-s − 5.98i·11-s + (0.707 − 0.707i)12-s + (0.515 + 0.515i)13-s − 1.60i·14-s + (−1.64 − 1.51i)15-s − 1.00·16-s + (2.59 − 2.59i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.408 − 0.408i)3-s + 0.500i·4-s + (0.999 − 0.0431i)5-s − 0.408i·6-s + (−0.429 − 0.429i)7-s + (−0.250 + 0.250i)8-s + 0.333i·9-s + (0.521 + 0.477i)10-s − 1.80i·11-s + (0.204 − 0.204i)12-s + (0.142 + 0.142i)13-s − 0.429i·14-s + (−0.425 − 0.390i)15-s − 0.250·16-s + (0.629 − 0.629i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 + 0.532i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.846 + 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.846 + 0.532i$
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: \((2,\ 930,\ (\ :1/2),\ 0.846 + 0.532i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.89044 - 0.544992i\)
\(L(\frac12)\) \(\approx\) \(1.89044 - 0.544992i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 + 0.0964i)T \)
31 \( 1 + (-5.10 + 2.23i)T \)
good7 \( 1 + (1.13 + 1.13i)T + 7iT^{2} \)
11 \( 1 + 5.98iT - 11T^{2} \)
13 \( 1 + (-0.515 - 0.515i)T + 13iT^{2} \)
17 \( 1 + (-2.59 + 2.59i)T - 17iT^{2} \)
19 \( 1 + 0.796iT - 19T^{2} \)
23 \( 1 + (0.904 + 0.904i)T + 23iT^{2} \)
29 \( 1 + 3.11T + 29T^{2} \)
37 \( 1 + (2.27 - 2.27i)T - 37iT^{2} \)
41 \( 1 - 3.72T + 41T^{2} \)
43 \( 1 + (2.06 + 2.06i)T + 43iT^{2} \)
47 \( 1 + (-5.48 - 5.48i)T + 47iT^{2} \)
53 \( 1 + (-5.23 - 5.23i)T + 53iT^{2} \)
59 \( 1 + 9.53iT - 59T^{2} \)
61 \( 1 - 1.27iT - 61T^{2} \)
67 \( 1 + (4.19 + 4.19i)T + 67iT^{2} \)
71 \( 1 + 6.20T + 71T^{2} \)
73 \( 1 + (-10.1 - 10.1i)T + 73iT^{2} \)
79 \( 1 - 1.08T + 79T^{2} \)
83 \( 1 + (-5.43 - 5.43i)T + 83iT^{2} \)
89 \( 1 + 0.818T + 89T^{2} \)
97 \( 1 + (13.3 + 13.3i)T + 97iT^{2} \)
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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

−10.01753555872139567523002303469, −9.064526850780846546770992369718, −8.227600252530803511918000159642, −7.21812985164284082967446870557, −6.31534271409310063460084440793, −5.85140063726926633719233453809, −5.00872540271267344623180148730, −3.64124868760851324507615606472, −2.61517157577830625301625330978, −0.879193459017934062220885557545, 1.59630552841897039551357403263, 2.63122785771421847551097022500, 3.89925215782863677826501473895, 4.90444830235168147161837674449, 5.67690173031088858739300181149, 6.41360273237242654592593514513, 7.39961012795517697574765619428, 8.846192522695576182339752598811, 9.679855665744457272137786216984, 10.11348363882667041984185211369

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