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Properties

Label 2-930-31.25-c1-0-11
Degree $2$
Conductor $930$
Sign $0.992 - 0.122i$
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 + (2.11 + 3.66i)7-s + 8-s + (−0.499 + 0.866i)9-s + (0.5 − 0.866i)10-s + (−0.118 + 0.204i)11-s + (−0.5 − 0.866i)12-s + (2.11 + 3.66i)14-s − 0.999·15-s + 16-s + (3.85 + 6.67i)17-s + (−0.499 + 0.866i)18-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 + (0.800 + 1.38i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.158 − 0.273i)10-s + (−0.0355 + 0.0616i)11-s + (−0.144 − 0.249i)12-s + (0.566 + 0.980i)14-s − 0.258·15-s + 0.250·16-s + (0.934 + 1.61i)17-s + (−0.117 + 0.204i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.122i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.992 - 0.122i$
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: \((2,\ 930,\ (\ :1/2),\ 0.992 - 0.122i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.53905 + 0.156595i\)
\(L(\frac12)\) \(\approx\) \(2.53905 + 0.156595i\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.35 + 1.52i)T \)
good7 \( 1 + (-2.11 - 3.66i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.118 - 0.204i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.85 - 6.67i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.618 + 1.07i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.23T + 23T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
37 \( 1 + (1.61 + 2.80i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.61 + 2.80i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.23 + 9.06i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 5.23T + 47T^{2} \)
53 \( 1 + (0.736 - 1.27i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.881 - 1.52i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 8.18T + 61T^{2} \)
67 \( 1 + (0.381 - 0.661i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.23 - 5.60i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7.47 + 12.9i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.763 - 1.32i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.35 + 7.54i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 6.94T + 89T^{2} \)
97 \( 1 + 7.47T + 97T^{2} \)
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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

−10.31798951339799286152808702356, −9.014099306832109776610621383408, −8.399232554051796061570685080415, −7.52390583807641425799576811609, −6.38460600545476256265671725410, −5.55422950286914298306746318275, −5.15170737149064395230527577847, −3.84030815037765387422081616715, −2.43160366062712264929957063810, −1.54906714631110005559519872812, 1.16267744831252643065786330049, 2.87621108331005695076280069081, 3.84052534920316487701415817751, 4.79984101141725621638775107715, 5.40105213415700215547840144585, 6.67603705288710207954175819273, 7.30417471910208438713883780735, 8.162941304193230487656554828545, 9.543827850732447566007095368954, 10.19823009823766744315957559512

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