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Properties

Label 2-930-155.129-c1-0-7
Degree $2$
Conductor $930$
Sign $0.953 + 0.300i$
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 + (−0.968 + 2.01i)5-s + (−0.5 + 0.866i)6-s + (−3.61 − 2.08i)7-s + i·8-s + (0.499 + 0.866i)9-s + (2.01 + 0.968i)10-s + (−0.324 − 0.561i)11-s + (0.866 + 0.5i)12-s + (1.99 − 1.15i)13-s + (−2.08 + 3.61i)14-s + (1.84 − 1.26i)15-s + 16-s + (0.894 + 0.516i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.499 − 0.288i)3-s − 0.5·4-s + (−0.433 + 0.901i)5-s + (−0.204 + 0.353i)6-s + (−1.36 − 0.789i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (0.637 + 0.306i)10-s + (−0.0977 − 0.169i)11-s + (0.249 + 0.144i)12-s + (0.553 − 0.319i)13-s + (−0.558 + 0.967i)14-s + (0.476 − 0.325i)15-s + 0.250·16-s + (0.217 + 0.125i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 + 0.300i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.953 + 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.953 + 0.300i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.953 + 0.300i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.823821 - 0.126620i\)
\(L(\frac12)\) \(\approx\) \(0.823821 - 0.126620i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.968 - 2.01i)T \)
31 \( 1 + (-2.68 + 4.87i)T \)
good7 \( 1 + (3.61 + 2.08i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.324 + 0.561i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.99 + 1.15i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.894 - 0.516i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.12 - 3.68i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.34iT - 23T^{2} \)
29 \( 1 - 9.24T + 29T^{2} \)
37 \( 1 + (-9.55 - 5.51i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.04 - 3.54i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.09 - 2.94i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.42iT - 47T^{2} \)
53 \( 1 + (-5.52 + 3.19i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.574 + 0.995i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 8.11T + 61T^{2} \)
67 \( 1 + (9.53 - 5.50i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.71 + 2.96i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-7.38 + 4.26i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.47 - 14.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.43 + 1.40i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 13.8T + 89T^{2} \)
97 \( 1 + 0.0488iT - 97T^{2} \)
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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

−10.20437682114583831610932013141, −9.633365527476864323278691939301, −8.230460393277535437381965500426, −7.50572949827690243812162216483, −6.41919992592149139844309640132, −6.00917301672150729675468210025, −4.36875986530176581551407323607, −3.53558600935549788461323557550, −2.71065655837370288644574415958, −0.885782600196161307798856925786, 0.60759009027784497338106601681, 2.82839321061515683218738309029, 4.14590649330716206155594115081, 4.86073809556299566679044131003, 5.96004490973227270880763418443, 6.44002814259045763928369206032, 7.49047284341318253867427784413, 8.761076315425341206336765813290, 8.966486034329832572420629324361, 9.914472102247121727429425496887

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