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Properties

Label 2-930-15.2-c1-0-3
Degree $2$
Conductor $930$
Sign $0.278 - 0.960i$
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 + (−0.292 − 1.70i)3-s − 1.00i·4-s + (−0.489 + 2.18i)5-s + (−1.41 − 0.999i)6-s + (−0.474 − 0.474i)7-s + (−0.707 − 0.707i)8-s + (−2.82 + i)9-s + (1.19 + 1.88i)10-s + 4.08i·11-s + (−1.70 + 0.292i)12-s + (−3.10 + 3.10i)13-s − 0.671·14-s + (3.86 + 0.196i)15-s − 1.00·16-s + (−4.06 + 4.06i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.169 − 0.985i)3-s − 0.500i·4-s + (−0.218 + 0.975i)5-s + (−0.577 − 0.408i)6-s + (−0.179 − 0.179i)7-s + (−0.250 − 0.250i)8-s + (−0.942 + 0.333i)9-s + (0.378 + 0.597i)10-s + 1.23i·11-s + (−0.492 + 0.0845i)12-s + (−0.861 + 0.861i)13-s − 0.179·14-s + (0.998 + 0.0507i)15-s − 0.250·16-s + (−0.985 + 0.985i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.278 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.278 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.278 - 0.960i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (497, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.278 - 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.571907 + 0.429457i\)
\(L(\frac12)\) \(\approx\) \(0.571907 + 0.429457i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.292 + 1.70i)T \)
5 \( 1 + (0.489 - 2.18i)T \)
31 \( 1 - T \)
good7 \( 1 + (0.474 + 0.474i)T + 7iT^{2} \)
11 \( 1 - 4.08iT - 11T^{2} \)
13 \( 1 + (3.10 - 3.10i)T - 13iT^{2} \)
17 \( 1 + (4.06 - 4.06i)T - 17iT^{2} \)
19 \( 1 + 1.57iT - 19T^{2} \)
23 \( 1 + (1.06 + 1.06i)T + 23iT^{2} \)
29 \( 1 + 1.87T + 29T^{2} \)
37 \( 1 + (2 + 2i)T + 37iT^{2} \)
41 \( 1 - 7.26iT - 41T^{2} \)
43 \( 1 + (-1.16 + 1.16i)T - 43iT^{2} \)
47 \( 1 + (0.520 - 0.520i)T - 47iT^{2} \)
53 \( 1 + (2.48 + 2.48i)T + 53iT^{2} \)
59 \( 1 - 4.40T + 59T^{2} \)
61 \( 1 - 11.2T + 61T^{2} \)
67 \( 1 + (5.37 + 5.37i)T + 67iT^{2} \)
71 \( 1 - 4.42iT - 71T^{2} \)
73 \( 1 + (7.86 - 7.86i)T - 73iT^{2} \)
79 \( 1 - 9.12iT - 79T^{2} \)
83 \( 1 + (8.35 + 8.35i)T + 83iT^{2} \)
89 \( 1 - 5.69T + 89T^{2} \)
97 \( 1 + (-8.34 - 8.34i)T + 97iT^{2} \)
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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

−10.38065925707056542785814891930, −9.659762612144430905401963276751, −8.487295859475516452958426080178, −7.31777358561120146716565335177, −6.87030656021301716596957363684, −6.15008660348775559105891526187, −4.85071754308855974134270111697, −3.89459038795344976961591592598, −2.52876465515745990889822937007, −1.89743625596477068033474476882, 0.26941786097721548168154628210, 2.76425803474612312926511452961, 3.76251143522893554252333932214, 4.72621020712984179988137246974, 5.40680039015041144996047589666, 6.07305664488126583064772813964, 7.40902807999662342555545872806, 8.377404164023938352431786949111, 8.964227722967320513707883633321, 9.741226516539289495562994281776

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