Properties

Label 2-930-31.20-c1-0-6
Degree $2$
Conductor $930$
Sign $-0.935 - 0.352i$
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  + (0.309 + 0.951i)2-s + (0.669 + 0.743i)3-s + (−0.809 + 0.587i)4-s + (−0.5 − 0.866i)5-s + (−0.499 + 0.866i)6-s + (0.408 + 3.88i)7-s + (−0.809 − 0.587i)8-s + (−0.104 + 0.994i)9-s + (0.669 − 0.743i)10-s + (3.25 + 1.44i)11-s + (−0.978 − 0.207i)12-s + (−3.59 + 0.764i)13-s + (−3.56 + 1.58i)14-s + (0.309 − 0.951i)15-s + (0.309 − 0.951i)16-s + (−0.319 + 0.142i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (0.386 + 0.429i)3-s + (−0.404 + 0.293i)4-s + (−0.223 − 0.387i)5-s + (−0.204 + 0.353i)6-s + (0.154 + 1.46i)7-s + (−0.286 − 0.207i)8-s + (−0.0348 + 0.331i)9-s + (0.211 − 0.235i)10-s + (0.980 + 0.436i)11-s + (−0.282 − 0.0600i)12-s + (−0.998 + 0.212i)13-s + (−0.953 + 0.424i)14-s + (0.0797 − 0.245i)15-s + (0.0772 − 0.237i)16-s + (−0.0774 + 0.0344i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 - 0.352i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.935 - 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.935 - 0.352i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.935 - 0.352i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.281567 + 1.54480i\)
\(L(\frac12)\) \(\approx\) \(0.281567 + 1.54480i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
3 \( 1 + (-0.669 - 0.743i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (-4.71 + 2.96i)T \)
good7 \( 1 + (-0.408 - 3.88i)T + (-6.84 + 1.45i)T^{2} \)
11 \( 1 + (-3.25 - 1.44i)T + (7.36 + 8.17i)T^{2} \)
13 \( 1 + (3.59 - 0.764i)T + (11.8 - 5.28i)T^{2} \)
17 \( 1 + (0.319 - 0.142i)T + (11.3 - 12.6i)T^{2} \)
19 \( 1 + (2.66 + 0.566i)T + (17.3 + 7.72i)T^{2} \)
23 \( 1 + (-1.27 - 0.928i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-0.283 - 0.872i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 + (3.13 - 5.43i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.39 - 5.99i)T + (-4.28 - 40.7i)T^{2} \)
43 \( 1 + (9.44 + 2.00i)T + (39.2 + 17.4i)T^{2} \)
47 \( 1 + (1.30 - 4.01i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (0.146 - 1.39i)T + (-51.8 - 11.0i)T^{2} \)
59 \( 1 + (-6.16 - 6.84i)T + (-6.16 + 58.6i)T^{2} \)
61 \( 1 - 4.56T + 61T^{2} \)
67 \( 1 + (1.34 + 2.32i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.746 + 7.10i)T + (-69.4 - 14.7i)T^{2} \)
73 \( 1 + (-0.290 - 0.129i)T + (48.8 + 54.2i)T^{2} \)
79 \( 1 + (-14.9 + 6.65i)T + (52.8 - 58.7i)T^{2} \)
83 \( 1 + (6.51 - 7.23i)T + (-8.67 - 82.5i)T^{2} \)
89 \( 1 + (-4.58 + 3.32i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-4.44 + 3.22i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.07042961636550673452157099550, −9.378069462591211239742154149506, −8.712382019702665775794739336039, −8.124195282153077242288877743512, −6.98226176517544753530870773791, −6.13594462361486447769533425767, −5.02304537768428812211746025149, −4.53054797745734228932872672894, −3.23720748465118876362026244720, −2.01634136575593176715223297548, 0.65568597197372363671241111734, 2.00900511437201687090552968646, 3.33739071346064971768848354139, 4.00626692582739164035086169125, 5.03567438680958847240021167660, 6.54424604373575543694440099559, 7.05741699066738693398211380219, 8.042698090507230659502403273494, 8.877297640522854822308154257024, 9.996145659601413677756984961791

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