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Properties

Label 2-930-155.129-c1-0-24
Degree $2$
Conductor $930$
Sign $-0.292 + 0.956i$
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 + (0.866 + 0.5i)3-s − 4-s + (−1.82 + 1.28i)5-s + (0.5 − 0.866i)6-s + (−2.60 − 1.50i)7-s + i·8-s + (0.499 + 0.866i)9-s + (1.28 + 1.82i)10-s + (0.600 + 1.04i)11-s + (−0.866 − 0.5i)12-s + (4.95 − 2.85i)13-s + (−1.50 + 2.60i)14-s + (−2.22 + 0.202i)15-s + 16-s + (−2.42 − 1.40i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.499 + 0.288i)3-s − 0.5·4-s + (−0.817 + 0.576i)5-s + (0.204 − 0.353i)6-s + (−0.984 − 0.568i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (0.407 + 0.577i)10-s + (0.181 + 0.313i)11-s + (−0.249 − 0.144i)12-s + (1.37 − 0.792i)13-s + (−0.402 + 0.696i)14-s + (−0.574 + 0.0522i)15-s + 0.250·16-s + (−0.588 − 0.339i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.292 + 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.292 + 0.956i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.292 + 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.682818 - 0.923038i\)
\(L(\frac12)\) \(\approx\) \(0.682818 - 0.923038i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (1.82 - 1.28i)T \)
31 \( 1 + (5.56 - 0.0753i)T \)
good7 \( 1 + (2.60 + 1.50i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.600 - 1.04i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.95 + 2.85i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.42 + 1.40i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.56 + 4.43i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 4.06iT - 23T^{2} \)
29 \( 1 - 8.23T + 29T^{2} \)
37 \( 1 + (9.68 + 5.59i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.98 + 6.89i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.96 + 2.28i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 1.45iT - 47T^{2} \)
53 \( 1 + (-10.7 + 6.23i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.11 - 8.86i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 15.2T + 61T^{2} \)
67 \( 1 + (-2.67 + 1.54i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.59 - 13.1i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.91 - 3.41i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.63 + 11.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.78 + 2.76i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 3.80T + 89T^{2} \)
97 \( 1 - 1.57iT - 97T^{2} \)
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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

−10.15922660907196360983401190772, −8.897567668983020499532193394680, −8.488290708923059990252683313748, −7.19344883789916008974052484958, −6.67301906580849479704796260687, −5.18520692094889080882047308058, −3.92943833390002385238823390978, −3.49429338437088343619528990207, −2.52491489532810652941013289497, −0.55779071225113229042989943452, 1.39851301354282110991701652735, 3.33470061546963140532502595540, 3.87277313202991757724138892545, 5.17699702174148988579609050054, 6.26737650880013180855615580365, 6.80557382761977848222233985167, 7.972174800964800252409889935918, 8.587548104254350743065492807276, 9.106293953191294382433924189999, 10.00213201612721151389236157402

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