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Properties

Label 2-930-155.149-c1-0-13
Degree $2$
Conductor $930$
Sign $0.701 + 0.712i$
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 + (−1.86 + 1.23i)5-s + (−0.5 − 0.866i)6-s + (−2.29 + 1.32i)7-s − i·8-s + (0.499 − 0.866i)9-s + (−1.23 − 1.86i)10-s + (−2.32 + 4.02i)11-s + (0.866 − 0.5i)12-s + (−4.58 − 2.64i)13-s + (−1.32 − 2.29i)14-s + (1 − 2i)15-s + 16-s + (5.19 − 3i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.499 + 0.288i)3-s − 0.5·4-s + (−0.834 + 0.550i)5-s + (−0.204 − 0.353i)6-s + (−0.866 + 0.499i)7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.389 − 0.590i)10-s + (−0.700 + 1.21i)11-s + (0.249 − 0.144i)12-s + (−1.27 − 0.733i)13-s + (−0.353 − 0.612i)14-s + (0.258 − 0.516i)15-s + 0.250·16-s + (1.26 − 0.727i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.701 + 0.712i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.701 + 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.160239 - 0.0671241i\)
\(L(\frac12)\) \(\approx\) \(0.160239 - 0.0671241i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.86 - 1.23i)T \)
31 \( 1 + (4.14 + 3.71i)T \)
good7 \( 1 + (2.29 - 1.32i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.32 - 4.02i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.58 + 2.64i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-5.19 + 3i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.64 - 4.58i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 1.29iT - 23T^{2} \)
29 \( 1 - 2.64T + 29T^{2} \)
37 \( 1 + (3.46 - 2i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.64 + 6.31i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.613 + 0.354i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.29iT - 47T^{2} \)
53 \( 1 + (7.18 + 4.14i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.32 - 4.02i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 + (0.613 + 0.354i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.29 + 12.6i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (12.6 + 7.29i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.29 + 5.70i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-10.0 - 5.79i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 7.29T + 89T^{2} \)
97 \( 1 - 15.2iT - 97T^{2} \)
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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

−10.01989513361187903127256988726, −9.300843822635289210638342557230, −7.80293581657854302539920793518, −7.54453000760437295778930184463, −6.62707107827844541678096539015, −5.53302334565759812498615252697, −4.93140192899699944476154371329, −3.68999635571655246625337107915, −2.70370869692462888955439434653, −0.10627728782156412516692214117, 1.04832364313288995846244089579, 2.86979760780714482564938226592, 3.72573429974565983138568579659, 4.85267642438165542589573001766, 5.62153268662120148283241266260, 6.90683352906252592184324029292, 7.65696723929397351410220427468, 8.543616782108560005700778706838, 9.525053776113565914782966975012, 10.23184110942000027376290647173

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