Properties

Label 2-930-155.149-c1-0-4
Degree $2$
Conductor $930$
Sign $-0.754 - 0.655i$
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 + (−1.89 − 1.19i)5-s + (−0.5 − 0.866i)6-s + (2.10 − 1.21i)7-s i·8-s + (0.499 − 0.866i)9-s + (1.19 − 1.89i)10-s + (−0.345 + 0.598i)11-s + (0.866 − 0.5i)12-s + (−1.66 − 0.959i)13-s + (1.21 + 2.10i)14-s + (2.23 + 0.0874i)15-s + 16-s + (−4.39 + 2.53i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.499 + 0.288i)3-s − 0.5·4-s + (−0.845 − 0.533i)5-s + (−0.204 − 0.353i)6-s + (0.796 − 0.460i)7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (0.377 − 0.598i)10-s + (−0.104 + 0.180i)11-s + (0.249 − 0.144i)12-s + (−0.460 − 0.266i)13-s + (0.325 + 0.563i)14-s + (0.576 + 0.0225i)15-s + 0.250·16-s + (−1.06 + 0.616i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.754 - 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.754 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.754 - 0.655i$
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: \((2,\ 930,\ (\ :1/2),\ -0.754 - 0.655i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.258117 + 0.690455i\)
\(L(\frac12)\) \(\approx\) \(0.258117 + 0.690455i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.89 + 1.19i)T \)
31 \( 1 + (-4.90 + 2.63i)T \)
good7 \( 1 + (-2.10 + 1.21i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.345 - 0.598i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.66 + 0.959i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (4.39 - 2.53i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.14 - 3.71i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 6.84iT - 23T^{2} \)
29 \( 1 - 6.99T + 29T^{2} \)
37 \( 1 + (-1.27 + 0.735i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.94 - 10.2i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (9.13 - 5.27i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.99iT - 47T^{2} \)
53 \( 1 + (9.46 + 5.46i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.78 - 3.09i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 7.77T + 61T^{2} \)
67 \( 1 + (-12.2 - 7.09i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.66 - 9.80i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.34 - 3.08i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.37 + 12.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.756 + 0.436i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 7.31T + 89T^{2} \)
97 \( 1 - 9.79iT - 97T^{2} \)
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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

−10.26748328507959532083638067238, −9.581251543858150451769331521499, −8.338259553447905281136309319174, −7.990605046897914878735237068517, −7.06793249126183697406818827698, −6.08457631154323227378872368540, −4.90771788992955894443838814919, −4.57138619556375464907314480100, −3.46142501926066953045393324821, −1.30527569639783466997407858620, 0.41018318466357605305200130295, 2.15455869239395411832857193301, 3.08512359439106773956663899602, 4.61306940931807395428755247980, 4.90724091236908953463868728078, 6.46466404823517478959517554823, 7.12840631231530663159442908225, 8.257532080418424246600603430846, 8.767086679390973552451535969775, 10.01306787300326652448916414133

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