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Properties

Label 2-930-155.149-c1-0-17
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: $\chi_{930} (769, \cdot )$
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

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

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