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Properties

Label 2-930-155.129-c1-0-22
Degree $2$
Conductor $930$
Sign $0.807 + 0.589i$
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 + (1.91 + 1.14i)5-s + (0.5 − 0.866i)6-s + (−0.230 − 0.133i)7-s + i·8-s + (0.499 + 0.866i)9-s + (1.14 − 1.91i)10-s + (−2.16 − 3.74i)11-s + (−0.866 − 0.5i)12-s + (3.19 − 1.84i)13-s + (−0.133 + 0.230i)14-s + (1.08 + 1.95i)15-s + 16-s + (3.14 + 1.81i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.499 + 0.288i)3-s − 0.5·4-s + (0.857 + 0.514i)5-s + (0.204 − 0.353i)6-s + (−0.0872 − 0.0503i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (0.363 − 0.606i)10-s + (−0.652 − 1.13i)11-s + (−0.249 − 0.144i)12-s + (0.885 − 0.511i)13-s + (−0.0356 + 0.0616i)14-s + (0.280 + 0.504i)15-s + 0.250·16-s + (0.762 + 0.440i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.807 + 0.589i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.807 + 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.807 + 0.589i$
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: \((2,\ 930,\ (\ :1/2),\ 0.807 + 0.589i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.01081 - 0.655611i\)
\(L(\frac12)\) \(\approx\) \(2.01081 - 0.655611i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.91 - 1.14i)T \)
31 \( 1 + (4.12 + 3.73i)T \)
good7 \( 1 + (0.230 + 0.133i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.16 + 3.74i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.19 + 1.84i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.14 - 1.81i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.627 + 1.08i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 2.23iT - 23T^{2} \)
29 \( 1 - 8.58T + 29T^{2} \)
37 \( 1 + (-5.89 - 3.40i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.09 - 5.36i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-8.46 - 4.88i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 9.03iT - 47T^{2} \)
53 \( 1 + (6.70 - 3.87i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.855 + 1.48i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 5.59T + 61T^{2} \)
67 \( 1 + (1.14 - 0.658i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.22 + 9.04i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.38 + 3.10i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.29 + 3.97i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (9.52 - 5.49i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 13.8T + 89T^{2} \)
97 \( 1 + 11.1iT - 97T^{2} \)
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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

−10.02333195518754674700901249101, −9.370928615469657352158815631370, −8.404558733311926430339240952605, −7.79579817798868100078590787665, −6.27025539778407160376582967159, −5.70271831526279133785525684338, −4.47606673033681698186654065698, −3.17234136470932761618979621128, −2.77506717948495227915462381957, −1.20986855539327981038681795267, 1.32250741974182438945071400023, 2.58205211901607037190131937209, 4.01491053089523940031540485938, 5.06591162061787965391431656708, 5.85022253108575624135766821594, 6.81663444149570660121719153580, 7.59520265326118252367837769996, 8.466697511778478398675190311106, 9.245847073055343872374568833918, 9.832911401456599338292830706850

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