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Properties

Label 2-930-465.464-c1-0-25
Degree $2$
Conductor $930$
Sign $0.165 - 0.986i$
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 + (−0.522 + 1.65i)3-s + 4-s + (1.93 − 1.11i)5-s + (−0.522 + 1.65i)6-s + 3.41i·7-s + 8-s + (−2.45 − 1.72i)9-s + (1.93 − 1.11i)10-s + 3.28·11-s + (−0.522 + 1.65i)12-s − 4.68·13-s + 3.41i·14-s + (0.834 + 3.78i)15-s + 16-s + 5.44i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.301 + 0.953i)3-s + 0.5·4-s + (0.866 − 0.500i)5-s + (−0.213 + 0.674i)6-s + 1.29i·7-s + 0.353·8-s + (−0.817 − 0.575i)9-s + (0.612 − 0.353i)10-s + 0.991·11-s + (−0.150 + 0.476i)12-s − 1.29·13-s + 0.913i·14-s + (0.215 + 0.976i)15-s + 0.250·16-s + 1.32i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.165 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.165 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.165 - 0.986i$
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: \((2,\ 930,\ (\ :1/2),\ 0.165 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90877 + 1.61558i\)
\(L(\frac12)\) \(\approx\) \(1.90877 + 1.61558i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.522 - 1.65i)T \)
5 \( 1 + (-1.93 + 1.11i)T \)
31 \( 1 + (5.16 + 2.08i)T \)
good7 \( 1 - 3.41iT - 7T^{2} \)
11 \( 1 - 3.28T + 11T^{2} \)
13 \( 1 + 4.68T + 13T^{2} \)
17 \( 1 - 5.44iT - 17T^{2} \)
19 \( 1 - 6.86T + 19T^{2} \)
23 \( 1 - 3.10iT - 23T^{2} \)
29 \( 1 + 6.42T + 29T^{2} \)
37 \( 1 - 9.62T + 37T^{2} \)
41 \( 1 - 2.33iT - 41T^{2} \)
43 \( 1 - 3.46T + 43T^{2} \)
47 \( 1 + 8.08T + 47T^{2} \)
53 \( 1 + 6.10iT - 53T^{2} \)
59 \( 1 - 10.9iT - 59T^{2} \)
61 \( 1 + 4.84iT - 61T^{2} \)
67 \( 1 + 1.08iT - 67T^{2} \)
71 \( 1 + 1.09iT - 71T^{2} \)
73 \( 1 + 2.96T + 73T^{2} \)
79 \( 1 + 1.97iT - 79T^{2} \)
83 \( 1 + 12.0iT - 83T^{2} \)
89 \( 1 - 7.11T + 89T^{2} \)
97 \( 1 - 14.3iT - 97T^{2} \)
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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

−10.06174688468844400332521723856, −9.441870418958298471700798210870, −9.003906968439042704771370881751, −7.72153678235676132632489679700, −6.33064765908257729343384201915, −5.65738738328857605101395782092, −5.17918315825274155523753356815, −4.13466359636346716712932903667, −3.00572625620100060565574525722, −1.80882701591660824930436443681, 1.03382399588111142169183045959, 2.31661953302100773034769152377, 3.35831839583420592723251071485, 4.73975103942491996123102531579, 5.56262355302310384930478231985, 6.56022069641722787454852676095, 7.29196279693769302223310678199, 7.51118645470392996976593079357, 9.305944583195825096918968003196, 9.894829527070301212129757472958

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