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Properties

Label 2-930-465.464-c1-0-34
Degree $2$
Conductor $930$
Sign $0.252 - 0.967i$
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  + 2-s + (1.02 + 1.39i)3-s + 4-s + (−0.467 + 2.18i)5-s + (1.02 + 1.39i)6-s − 1.43i·7-s + 8-s + (−0.884 + 2.86i)9-s + (−0.467 + 2.18i)10-s + 6.14·11-s + (1.02 + 1.39i)12-s + 2.15·13-s − 1.43i·14-s + (−3.52 + 1.59i)15-s + 16-s − 0.937i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.593 + 0.804i)3-s + 0.5·4-s + (−0.209 + 0.977i)5-s + (0.419 + 0.568i)6-s − 0.540i·7-s + 0.353·8-s + (−0.294 + 0.955i)9-s + (−0.147 + 0.691i)10-s + 1.85·11-s + (0.296 + 0.402i)12-s + 0.597·13-s − 0.382i·14-s + (−0.910 + 0.412i)15-s + 0.250·16-s − 0.227i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.252 - 0.967i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (929, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.252 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.40577 + 1.85821i\)
\(L(\frac12)\) \(\approx\) \(2.40577 + 1.85821i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-1.02 - 1.39i)T \)
5 \( 1 + (0.467 - 2.18i)T \)
31 \( 1 + (3.50 - 4.32i)T \)
good7 \( 1 + 1.43iT - 7T^{2} \)
11 \( 1 - 6.14T + 11T^{2} \)
13 \( 1 - 2.15T + 13T^{2} \)
17 \( 1 + 0.937iT - 17T^{2} \)
19 \( 1 + 5.64T + 19T^{2} \)
23 \( 1 + 0.266iT - 23T^{2} \)
29 \( 1 + 3.72T + 29T^{2} \)
37 \( 1 - 7.85T + 37T^{2} \)
41 \( 1 - 3.31iT - 41T^{2} \)
43 \( 1 - 0.581T + 43T^{2} \)
47 \( 1 + 4.74T + 47T^{2} \)
53 \( 1 + 6.07iT - 53T^{2} \)
59 \( 1 - 5.39iT - 59T^{2} \)
61 \( 1 + 9.71iT - 61T^{2} \)
67 \( 1 + 9.90iT - 67T^{2} \)
71 \( 1 + 1.07iT - 71T^{2} \)
73 \( 1 + 3.14T + 73T^{2} \)
79 \( 1 + 2.07iT - 79T^{2} \)
83 \( 1 + 8.35iT - 83T^{2} \)
89 \( 1 - 6.18T + 89T^{2} \)
97 \( 1 + 10.6iT - 97T^{2} \)
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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

−10.37431685746623532049101758166, −9.487640377132161189798532874999, −8.642135835190437082169669137080, −7.60307697980445702056869043713, −6.69811101403429447003885259400, −6.03638664668205262237601955517, −4.55645531864538402944119625099, −3.87564866084326781591707030312, −3.27270171755271513428134784608, −1.91505390811686716090842971558, 1.22237819546427575388195895054, 2.21644233041286352164987752881, 3.74419797060573972746261623877, 4.24622274861224508247492154190, 5.76658283026868245367722894018, 6.29612473395310897214566213225, 7.28145515442013015502655337168, 8.318221117930773299426104836418, 8.914733366919737743773394223314, 9.535115051649442548644415536686

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