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Properties

Label 2-930-15.8-c1-0-5
Degree $2$
Conductor $930$
Sign $-0.749 + 0.662i$
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 + (0.292 + 1.70i)3-s + 1.00i·4-s + (−2.12 + 0.707i)5-s + (−0.999 + 1.41i)6-s + (−0.707 + 0.707i)8-s + (−2.82 + i)9-s + (−2 − 0.999i)10-s + 4.24i·11-s + (−1.70 + 0.292i)12-s + (−3 − 3i)13-s + (−1.82 − 3.41i)15-s − 1.00·16-s + (1.41 + 1.41i)17-s + (−2.70 − 1.29i)18-s − 4i·19-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.169 + 0.985i)3-s + 0.500i·4-s + (−0.948 + 0.316i)5-s + (−0.408 + 0.577i)6-s + (−0.250 + 0.250i)8-s + (−0.942 + 0.333i)9-s + (−0.632 − 0.316i)10-s + 1.27i·11-s + (−0.492 + 0.0845i)12-s + (−0.832 − 0.832i)13-s + (−0.472 − 0.881i)15-s − 0.250·16-s + (0.342 + 0.342i)17-s + (−0.638 − 0.304i)18-s − 0.917i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.749 + 0.662i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (683, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.749 + 0.662i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.337168 - 0.891041i\)
\(L(\frac12)\) \(\approx\) \(0.337168 - 0.891041i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.292 - 1.70i)T \)
5 \( 1 + (2.12 - 0.707i)T \)
31 \( 1 + T \)
good7 \( 1 - 7iT^{2} \)
11 \( 1 - 4.24iT - 11T^{2} \)
13 \( 1 + (3 + 3i)T + 13iT^{2} \)
17 \( 1 + (-1.41 - 1.41i)T + 17iT^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 - 23iT^{2} \)
29 \( 1 + 2.82T + 29T^{2} \)
37 \( 1 + (5 - 5i)T - 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (-8 - 8i)T + 43iT^{2} \)
47 \( 1 + (4.24 + 4.24i)T + 47iT^{2} \)
53 \( 1 + (4.24 - 4.24i)T - 53iT^{2} \)
59 \( 1 + 5.65T + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 + (1 - i)T - 67iT^{2} \)
71 \( 1 + 1.41iT - 71T^{2} \)
73 \( 1 + (-2 - 2i)T + 73iT^{2} \)
79 \( 1 - 10iT - 79T^{2} \)
83 \( 1 + (-12.7 + 12.7i)T - 83iT^{2} \)
89 \( 1 + 7.07T + 89T^{2} \)
97 \( 1 + (-3 + 3i)T - 97iT^{2} \)
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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

−10.56525388434946245156269644052, −9.755798799219257423638855381359, −8.894781933395100476992622776885, −7.80824044609798389571681304785, −7.41384087811808677296044834966, −6.24219707712050805445023914349, −4.98184243954086951435181798117, −4.56650918508809601958237593532, −3.50464013593630016629411799984, −2.62623386413265361043416781078, 0.35845225930331716462255307627, 1.78449513312604381026496152832, 3.10199337933578825288728933139, 3.89076450321693730876667080315, 5.15497109352716466242633866706, 6.02703724300821650857947394985, 7.08232546690759435107189515270, 7.79739742810154540762501862706, 8.675894411162824492333883841831, 9.407574252983029967659291355764

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