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Properties

Label 2-930-31.25-c1-0-10
Degree $2$
Conductor $930$
Sign $0.275 - 0.961i$
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  + 2-s + (0.5 + 0.866i)3-s + 4-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)6-s + (1.91 + 3.31i)7-s + 8-s + (−0.499 + 0.866i)9-s + (−0.5 + 0.866i)10-s + (1.91 − 3.31i)11-s + (0.5 + 0.866i)12-s + (−1 + 1.73i)13-s + (1.91 + 3.31i)14-s − 0.999·15-s + 16-s + (−1.82 − 3.16i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.288 + 0.499i)3-s + 0.5·4-s + (−0.223 + 0.387i)5-s + (0.204 + 0.353i)6-s + (0.723 + 1.25i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.158 + 0.273i)10-s + (0.577 − 0.999i)11-s + (0.144 + 0.249i)12-s + (−0.277 + 0.480i)13-s + (0.511 + 0.886i)14-s − 0.258·15-s + 0.250·16-s + (−0.443 − 0.768i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.275 - 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.275 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.275 - 0.961i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.275 - 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.20198 + 1.66035i\)
\(L(\frac12)\) \(\approx\) \(2.20198 + 1.66035i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (3.5 - 4.33i)T \)
good7 \( 1 + (-1.91 - 3.31i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.91 + 3.31i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.82 + 3.16i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3 - 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 3.65T + 23T^{2} \)
29 \( 1 - 5.82T + 29T^{2} \)
37 \( 1 + (4.82 + 8.36i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.65 - 9.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.828 - 1.43i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 9.31T + 47T^{2} \)
53 \( 1 + (-6.32 + 10.9i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.91 - 5.04i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6 + 10.3i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.65 + 11.5i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.32 - 2.30i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 12T + 89T^{2} \)
97 \( 1 + 11.4T + 97T^{2} \)
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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

−10.37295633952254766341578900371, −9.305694427155223829868261412492, −8.595144713939553808462200646928, −7.79315245757739800126941780852, −6.64199057049906784210940044231, −5.71611330976889986228471935790, −5.00333511722406565367847323251, −3.90041585808802891451286500433, −3.00594743725426921109884920505, −1.92066459610761922666650533171, 1.09188113873534649490456895264, 2.28418655424084891227949085317, 3.78088894861765260221186342736, 4.43457742587136057054502253708, 5.33803651742809362580953128191, 6.69363605504370887591560494262, 7.25644729152691080445132938237, 7.974814412316565349536692764816, 8.928440769258359650172328906183, 10.08166097307428095456518343101

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