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Properties

Label 2-930-93.92-c1-0-13
Degree $2$
Conductor $930$
Sign $0.754 - 0.656i$
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  + i·2-s + (0.224 − 1.71i)3-s − 4-s + i·5-s + (1.71 + 0.224i)6-s − 2.17·7-s − i·8-s + (−2.89 − 0.770i)9-s − 10-s + 3.24·11-s + (−0.224 + 1.71i)12-s + 4.02i·13-s − 2.17i·14-s + (1.71 + 0.224i)15-s + 16-s − 1.33·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.129 − 0.991i)3-s − 0.5·4-s + 0.447i·5-s + (0.701 + 0.0915i)6-s − 0.820·7-s − 0.353i·8-s + (−0.966 − 0.256i)9-s − 0.316·10-s + 0.977·11-s + (−0.0647 + 0.495i)12-s + 1.11i·13-s − 0.580i·14-s + (0.443 + 0.0579i)15-s + 0.250·16-s − 0.324·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.754 - 0.656i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.754 - 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.754 - 0.656i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (371, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.754 - 0.656i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34857 + 0.504589i\)
\(L(\frac12)\) \(\approx\) \(1.34857 + 0.504589i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (-0.224 + 1.71i)T \)
5 \( 1 - iT \)
31 \( 1 + (-4.16 - 3.69i)T \)
good7 \( 1 + 2.17T + 7T^{2} \)
11 \( 1 - 3.24T + 11T^{2} \)
13 \( 1 - 4.02iT - 13T^{2} \)
17 \( 1 + 1.33T + 17T^{2} \)
19 \( 1 - 5.97T + 19T^{2} \)
23 \( 1 - 6.67T + 23T^{2} \)
29 \( 1 - 4.69T + 29T^{2} \)
37 \( 1 - 3.43iT - 37T^{2} \)
41 \( 1 + 5.45iT - 41T^{2} \)
43 \( 1 + 7.95iT - 43T^{2} \)
47 \( 1 - 11.1iT - 47T^{2} \)
53 \( 1 - 11.9T + 53T^{2} \)
59 \( 1 + 12.6iT - 59T^{2} \)
61 \( 1 + 3.87iT - 61T^{2} \)
67 \( 1 + 3.59T + 67T^{2} \)
71 \( 1 - 8.31iT - 71T^{2} \)
73 \( 1 - 14.2iT - 73T^{2} \)
79 \( 1 - 12.4iT - 79T^{2} \)
83 \( 1 - 0.448T + 83T^{2} \)
89 \( 1 - 5.85T + 89T^{2} \)
97 \( 1 - 14.7T + 97T^{2} \)
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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

−9.849380745939345419880043668403, −9.139981439841487363495966147822, −8.479210432904095570580347084156, −7.24058053237242456053158013419, −6.81908039785805976197691575485, −6.29089935035670255762924675978, −5.13426237661788726815743201472, −3.76542844722922541295878423557, −2.76110096857915740611988419131, −1.14487671987483780490306549671, 0.854881806021379094059814245887, 2.81583754794717821356133709041, 3.46193368260562638757422386723, 4.50759921172428988708596390779, 5.34489805588683859512826091195, 6.32052013128317811388290175857, 7.66478795763710861157276349953, 8.758514732131164900430768901854, 9.262098955789477244911210341673, 9.982745727254692001036426475739

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