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Properties

Label 2-930-155.123-c1-0-19
Degree $2$
Conductor $930$
Sign $0.0403 + 0.999i$
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 + (−1.03 + 1.98i)5-s − 1.00i·6-s + (0.181 − 0.181i)7-s + (−0.707 − 0.707i)8-s − 1.00i·9-s + (0.669 + 2.13i)10-s − 2.46i·11-s + (−0.707 − 0.707i)12-s + (2.77 − 2.77i)13-s − 0.256i·14-s + (0.669 + 2.13i)15-s − 1.00·16-s + (3.68 + 3.68i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.408 − 0.408i)3-s − 0.500i·4-s + (−0.463 + 0.886i)5-s − 0.408i·6-s + (0.0684 − 0.0684i)7-s + (−0.250 − 0.250i)8-s − 0.333i·9-s + (0.211 + 0.674i)10-s − 0.744i·11-s + (−0.204 − 0.204i)12-s + (0.769 − 0.769i)13-s − 0.0684i·14-s + (0.172 + 0.550i)15-s − 0.250·16-s + (0.894 + 0.894i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0403 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0403 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.0403 + 0.999i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.0403 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59238 - 1.52945i\)
\(L(\frac12)\) \(\approx\) \(1.59238 - 1.52945i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.03 - 1.98i)T \)
31 \( 1 + (4.89 + 2.64i)T \)
good7 \( 1 + (-0.181 + 0.181i)T - 7iT^{2} \)
11 \( 1 + 2.46iT - 11T^{2} \)
13 \( 1 + (-2.77 + 2.77i)T - 13iT^{2} \)
17 \( 1 + (-3.68 - 3.68i)T + 17iT^{2} \)
19 \( 1 + 5.69iT - 19T^{2} \)
23 \( 1 + (-4.12 + 4.12i)T - 23iT^{2} \)
29 \( 1 - 3.69T + 29T^{2} \)
37 \( 1 + (1.14 + 1.14i)T + 37iT^{2} \)
41 \( 1 - 5.13T + 41T^{2} \)
43 \( 1 + (0.169 - 0.169i)T - 43iT^{2} \)
47 \( 1 + (4.17 - 4.17i)T - 47iT^{2} \)
53 \( 1 + (-8.45 + 8.45i)T - 53iT^{2} \)
59 \( 1 - 4.83iT - 59T^{2} \)
61 \( 1 - 8.97iT - 61T^{2} \)
67 \( 1 + (-2.21 + 2.21i)T - 67iT^{2} \)
71 \( 1 + 11.4T + 71T^{2} \)
73 \( 1 + (3.31 - 3.31i)T - 73iT^{2} \)
79 \( 1 - 9.78T + 79T^{2} \)
83 \( 1 + (7.11 - 7.11i)T - 83iT^{2} \)
89 \( 1 + 8.02T + 89T^{2} \)
97 \( 1 + (8.14 - 8.14i)T - 97iT^{2} \)
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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

−10.15735355214543806360451218093, −8.933603337996797708233179268885, −8.201470464070972726283746870249, −7.28309795222949483573916207432, −6.38839543042183859806127374008, −5.58840885950271499858498834916, −4.19606040655690359751542720199, −3.27554193019058881833874233500, −2.62121979060795232256883949587, −0.939045240216478914488161224115, 1.58955516925292374377961183001, 3.29174601791179578265604269688, 4.08360855000022859072913382062, 4.97045253821194241171966360717, 5.69409610718399578524572674948, 7.02116856036014878203514136472, 7.73077183056787096412395394613, 8.576672420839107413268710042327, 9.257841866819837564812981060989, 10.05830451448269928806294579995

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