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Properties

Label 2-930-465.464-c1-0-38
Degree $2$
Conductor $930$
Sign $0.939 - 0.341i$
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 + (1.63 + 0.571i)3-s + 4-s + (−1.46 − 1.69i)5-s + (1.63 + 0.571i)6-s + 2.83i·7-s + 8-s + (2.34 + 1.86i)9-s + (−1.46 − 1.69i)10-s + 3.85·11-s + (1.63 + 0.571i)12-s − 2.42·13-s + 2.83i·14-s + (−1.42 − 3.60i)15-s + 16-s − 4.97i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.943 + 0.330i)3-s + 0.5·4-s + (−0.654 − 0.756i)5-s + (0.667 + 0.233i)6-s + 1.07i·7-s + 0.353·8-s + (0.782 + 0.623i)9-s + (−0.462 − 0.534i)10-s + 1.16·11-s + (0.471 + 0.165i)12-s − 0.672·13-s + 0.758i·14-s + (−0.368 − 0.929i)15-s + 0.250·16-s − 1.20i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.341i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.939 - 0.341i$
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: \((2,\ 930,\ (\ :1/2),\ 0.939 - 0.341i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.17146 + 0.558998i\)
\(L(\frac12)\) \(\approx\) \(3.17146 + 0.558998i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.63 - 0.571i)T \)
5 \( 1 + (1.46 + 1.69i)T \)
31 \( 1 + (0.156 - 5.56i)T \)
good7 \( 1 - 2.83iT - 7T^{2} \)
11 \( 1 - 3.85T + 11T^{2} \)
13 \( 1 + 2.42T + 13T^{2} \)
17 \( 1 + 4.97iT - 17T^{2} \)
19 \( 1 - 8.42T + 19T^{2} \)
23 \( 1 - 0.144iT - 23T^{2} \)
29 \( 1 + 4.43T + 29T^{2} \)
37 \( 1 + 8.07T + 37T^{2} \)
41 \( 1 - 7.56iT - 41T^{2} \)
43 \( 1 - 0.244T + 43T^{2} \)
47 \( 1 + 0.441T + 47T^{2} \)
53 \( 1 + 9.87iT - 53T^{2} \)
59 \( 1 + 11.9iT - 59T^{2} \)
61 \( 1 + 10.1iT - 61T^{2} \)
67 \( 1 - 1.92iT - 67T^{2} \)
71 \( 1 + 0.737iT - 71T^{2} \)
73 \( 1 + 4.18T + 73T^{2} \)
79 \( 1 + 14.3iT - 79T^{2} \)
83 \( 1 - 4.53iT - 83T^{2} \)
89 \( 1 + 15.8T + 89T^{2} \)
97 \( 1 + 11.9iT - 97T^{2} \)
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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

−9.690514702826812918596263234959, −9.348496865025368590751386542708, −8.514164593941874694556646383995, −7.58769376896790071281441927607, −6.85247137947554018306168472335, −5.27345605511010031096607698008, −4.92100917836502744181450873197, −3.67580715668744903049476689954, −2.97871809687068616201263586764, −1.60380227235199943384331785280, 1.37779998971044869839719780673, 2.81661678009506143099335803392, 3.86223906788486117259320215256, 4.08220200439966348631963282970, 5.76322497977779747749380784552, 7.07708874852620553791850514278, 7.15701277436879997087577315452, 8.047749937214251758064605945439, 9.214437913197810906450650613952, 10.09644523038393119824842045409

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