Properties

Label 2-930-93.11-c1-0-7
Degree $2$
Conductor $930$
Sign $-0.150 - 0.988i$
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  + (−0.951 + 0.309i)2-s + (−1.68 + 0.393i)3-s + (0.809 − 0.587i)4-s + (0.866 − 0.5i)5-s + (1.48 − 0.895i)6-s + (0.503 + 4.79i)7-s + (−0.587 + 0.809i)8-s + (2.69 − 1.32i)9-s + (−0.669 + 0.743i)10-s + (3.34 + 1.48i)11-s + (−1.13 + 1.30i)12-s + (0.191 + 0.900i)13-s + (−1.96 − 4.40i)14-s + (−1.26 + 1.18i)15-s + (0.309 − 0.951i)16-s + (3.22 − 1.43i)17-s + ⋯
L(s)  = 1  + (−0.672 + 0.218i)2-s + (−0.973 + 0.227i)3-s + (0.404 − 0.293i)4-s + (0.387 − 0.223i)5-s + (0.605 − 0.365i)6-s + (0.190 + 1.81i)7-s + (−0.207 + 0.286i)8-s + (0.896 − 0.442i)9-s + (−0.211 + 0.235i)10-s + (1.00 + 0.449i)11-s + (−0.327 + 0.378i)12-s + (0.0530 + 0.249i)13-s + (−0.523 − 1.17i)14-s + (−0.326 + 0.305i)15-s + (0.0772 − 0.237i)16-s + (0.782 − 0.348i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.150 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.150 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.150 - 0.988i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.150 - 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.599291 + 0.697163i\)
\(L(\frac12)\) \(\approx\) \(0.599291 + 0.697163i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.951 - 0.309i)T \)
3 \( 1 + (1.68 - 0.393i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 + (-4.68 + 3.00i)T \)
good7 \( 1 + (-0.503 - 4.79i)T + (-6.84 + 1.45i)T^{2} \)
11 \( 1 + (-3.34 - 1.48i)T + (7.36 + 8.17i)T^{2} \)
13 \( 1 + (-0.191 - 0.900i)T + (-11.8 + 5.28i)T^{2} \)
17 \( 1 + (-3.22 + 1.43i)T + (11.3 - 12.6i)T^{2} \)
19 \( 1 + (2.91 + 0.618i)T + (17.3 + 7.72i)T^{2} \)
23 \( 1 + (0.0893 + 0.0649i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (1.02 + 3.14i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 + (0.546 + 0.315i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-7.90 - 7.12i)T + (4.28 + 40.7i)T^{2} \)
43 \( 1 + (2.30 - 10.8i)T + (-39.2 - 17.4i)T^{2} \)
47 \( 1 + (-1.38 - 0.451i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-0.987 + 9.39i)T + (-51.8 - 11.0i)T^{2} \)
59 \( 1 + (-0.439 + 0.395i)T + (6.16 - 58.6i)T^{2} \)
61 \( 1 - 6.64iT - 61T^{2} \)
67 \( 1 + (-6.53 - 11.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (11.4 + 1.19i)T + (69.4 + 14.7i)T^{2} \)
73 \( 1 + (0.190 - 0.426i)T + (-48.8 - 54.2i)T^{2} \)
79 \( 1 + (-1.63 - 3.66i)T + (-52.8 + 58.7i)T^{2} \)
83 \( 1 + (6.02 - 6.69i)T + (-8.67 - 82.5i)T^{2} \)
89 \( 1 + (-8.73 + 6.34i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (9.20 - 6.68i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.971555546497153213231361697955, −9.535041108139086737214380177190, −8.835475595565984955358844475709, −7.88856721863778705954532631715, −6.60400741836737851642833251050, −6.07863059682691901061003007331, −5.32097161623027085562084285984, −4.33887620667953706885412077160, −2.54581495082104696631349573736, −1.34525405177197914659568807321, 0.71164504744202728854481120444, 1.62678675232267869409125092101, 3.53272016058603687456849925350, 4.37747463732786275358008542164, 5.71251959406891079842790847770, 6.62290377419762936455874506395, 7.19713512929328291710710991002, 8.016145150219845149162248820974, 9.158274949569599844542111382774, 10.27797219747060831691231414050

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