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Properties

Label 2-930-31.16-c1-0-8
Degree $2$
Conductor $930$
Sign $0.994 - 0.100i$
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  + (−0.809 + 0.587i)2-s + (−0.809 − 0.587i)3-s + (0.309 − 0.951i)4-s − 5-s + 6-s + (0.367 − 1.13i)7-s + (0.309 + 0.951i)8-s + (0.309 + 0.951i)9-s + (0.809 − 0.587i)10-s + (−0.867 + 2.66i)11-s + (−0.809 + 0.587i)12-s + (−2.01 − 1.46i)13-s + (0.367 + 1.13i)14-s + (0.809 + 0.587i)15-s + (−0.809 − 0.587i)16-s + (1.51 + 4.67i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (−0.467 − 0.339i)3-s + (0.154 − 0.475i)4-s − 0.447·5-s + 0.408·6-s + (0.138 − 0.427i)7-s + (0.109 + 0.336i)8-s + (0.103 + 0.317i)9-s + (0.255 − 0.185i)10-s + (−0.261 + 0.804i)11-s + (−0.233 + 0.169i)12-s + (−0.560 − 0.406i)13-s + (0.0981 + 0.302i)14-s + (0.208 + 0.151i)15-s + (−0.202 − 0.146i)16-s + (0.368 + 1.13i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.994 - 0.100i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.994 - 0.100i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.857436 + 0.0432202i\)
\(L(\frac12)\) \(\approx\) \(0.857436 + 0.0432202i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 - 0.587i)T \)
3 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + T \)
31 \( 1 + (4.20 - 3.64i)T \)
good7 \( 1 + (-0.367 + 1.13i)T + (-5.66 - 4.11i)T^{2} \)
11 \( 1 + (0.867 - 2.66i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (2.01 + 1.46i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.51 - 4.67i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-3.29 + 2.39i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-0.331 - 1.01i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-5.49 + 3.99i)T + (8.96 - 27.5i)T^{2} \)
37 \( 1 + 4.80T + 37T^{2} \)
41 \( 1 + (-10.0 + 7.32i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + (-8.77 + 6.37i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (7.84 + 5.69i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-1.28 - 3.96i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-10.3 - 7.53i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 - 10.3T + 61T^{2} \)
67 \( 1 - 4.07T + 67T^{2} \)
71 \( 1 + (-4.35 - 13.3i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-1.98 + 6.11i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (4.85 + 14.9i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-11.5 + 8.36i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (2.27 - 6.98i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (0.377 - 1.16i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.28326746929108221228114452121, −9.192795751050995654395757491411, −8.226733276813987323872337070375, −7.39454630992595609868084694766, −7.04012530147674528268571558524, −5.80329606174161055626148521778, −5.01249526319257492874184846872, −3.89402289630719787317551954826, −2.28919546546848874497022329369, −0.816019940188614360441176854965, 0.824095338247814709825079315231, 2.57146079879295263548379184266, 3.55807323536578905207752834085, 4.76731278181208516729567315216, 5.61189542092179223127555139544, 6.76921577056702253895294509643, 7.67982829817117320576992498686, 8.455330226149661527140150011746, 9.431381047575721556672842907681, 9.917367149479644274063125805913

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