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Properties

Label 2-930-155.92-c1-0-25
Degree $2$
Conductor $930$
Sign $-0.997 + 0.0706i$
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  + (−0.707 − 0.707i)2-s + (−0.707 − 0.707i)3-s + 1.00i·4-s + (1.06 + 1.96i)5-s + 1.00i·6-s + (−2.96 − 2.96i)7-s + (0.707 − 0.707i)8-s + 1.00i·9-s + (0.636 − 2.14i)10-s − 3.23i·11-s + (0.707 − 0.707i)12-s + (1.33 + 1.33i)13-s + 4.19i·14-s + (0.636 − 2.14i)15-s − 1.00·16-s + (2.89 − 2.89i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.408 − 0.408i)3-s + 0.500i·4-s + (0.476 + 0.879i)5-s + 0.408i·6-s + (−1.12 − 1.12i)7-s + (0.250 − 0.250i)8-s + 0.333i·9-s + (0.201 − 0.677i)10-s − 0.975i·11-s + (0.204 − 0.204i)12-s + (0.371 + 0.371i)13-s + 1.12i·14-s + (0.164 − 0.553i)15-s − 0.250·16-s + (0.700 − 0.700i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0706i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0706i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.997 + 0.0706i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.997 + 0.0706i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0156787 - 0.443020i\)
\(L(\frac12)\) \(\approx\) \(0.0156787 - 0.443020i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (-1.06 - 1.96i)T \)
31 \( 1 + (3.10 + 4.62i)T \)
good7 \( 1 + (2.96 + 2.96i)T + 7iT^{2} \)
11 \( 1 + 3.23iT - 11T^{2} \)
13 \( 1 + (-1.33 - 1.33i)T + 13iT^{2} \)
17 \( 1 + (-2.89 + 2.89i)T - 17iT^{2} \)
19 \( 1 - 3.68iT - 19T^{2} \)
23 \( 1 + (-1.35 - 1.35i)T + 23iT^{2} \)
29 \( 1 + 10.3T + 29T^{2} \)
37 \( 1 + (-6.67 + 6.67i)T - 37iT^{2} \)
41 \( 1 + 10.3T + 41T^{2} \)
43 \( 1 + (3.51 + 3.51i)T + 43iT^{2} \)
47 \( 1 + (7.51 + 7.51i)T + 47iT^{2} \)
53 \( 1 + (2.49 + 2.49i)T + 53iT^{2} \)
59 \( 1 + 0.694iT - 59T^{2} \)
61 \( 1 - 3.53iT - 61T^{2} \)
67 \( 1 + (6.09 + 6.09i)T + 67iT^{2} \)
71 \( 1 + 8.76T + 71T^{2} \)
73 \( 1 + (4.41 + 4.41i)T + 73iT^{2} \)
79 \( 1 - 14.6T + 79T^{2} \)
83 \( 1 + (8.30 + 8.30i)T + 83iT^{2} \)
89 \( 1 + 15.9T + 89T^{2} \)
97 \( 1 + (-5.33 - 5.33i)T + 97iT^{2} \)
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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

−9.851603533194514129810589196532, −9.109443246598974031521662189553, −7.74975096790275933396180643986, −7.18460951818212364650352141189, −6.34367783853572937403503284496, −5.57986096518813146316654385328, −3.73271510222137904709527673326, −3.25942948767461286308200200422, −1.76031369418231731377990006460, −0.25644771841122919637531517596, 1.61597486486745978235836064785, 3.12238788882495644284917213991, 4.60158268008093768506397882070, 5.47121251730832495941991180609, 6.06648302714037605809707688133, 6.93285881438927795250315050991, 8.185118376704100737195290879030, 8.962776153430289108605165431883, 9.629464614561888907409472790775, 10.01326230935427053107927704199

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