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Properties

Label 2-930-15.2-c1-0-13
Degree $2$
Conductor $930$
Sign $-0.211 - 0.977i$
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 + (1.73 − 0.00608i)3-s − 1.00i·4-s + (−2.02 + 0.956i)5-s + (−1.22 + 1.22i)6-s + (−1.11 − 1.11i)7-s + (0.707 + 0.707i)8-s + (2.99 − 0.0210i)9-s + (0.753 − 2.10i)10-s + 2.70i·11-s + (−0.00608 − 1.73i)12-s + (0.120 − 0.120i)13-s + 1.57·14-s + (−3.49 + 1.66i)15-s − 1.00·16-s + (0.852 − 0.852i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.999 − 0.00351i)3-s − 0.500i·4-s + (−0.903 + 0.427i)5-s + (−0.498 + 0.501i)6-s + (−0.419 − 0.419i)7-s + (0.250 + 0.250i)8-s + (0.999 − 0.00703i)9-s + (0.238 − 0.665i)10-s + 0.816i·11-s + (−0.00175 − 0.499i)12-s + (0.0334 − 0.0334i)13-s + 0.419·14-s + (−0.902 + 0.430i)15-s − 0.250·16-s + (0.206 − 0.206i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.211 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.211 - 0.977i$
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: \((2,\ 930,\ (\ :1/2),\ -0.211 - 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.794766 + 0.985589i\)
\(L(\frac12)\) \(\approx\) \(0.794766 + 0.985589i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 + 0.00608i)T \)
5 \( 1 + (2.02 - 0.956i)T \)
31 \( 1 - T \)
good7 \( 1 + (1.11 + 1.11i)T + 7iT^{2} \)
11 \( 1 - 2.70iT - 11T^{2} \)
13 \( 1 + (-0.120 + 0.120i)T - 13iT^{2} \)
17 \( 1 + (-0.852 + 0.852i)T - 17iT^{2} \)
19 \( 1 - 5.63iT - 19T^{2} \)
23 \( 1 + (-6.00 - 6.00i)T + 23iT^{2} \)
29 \( 1 + 9.37T + 29T^{2} \)
37 \( 1 + (-4.37 - 4.37i)T + 37iT^{2} \)
41 \( 1 - 12.3iT - 41T^{2} \)
43 \( 1 + (6.77 - 6.77i)T - 43iT^{2} \)
47 \( 1 + (1.75 - 1.75i)T - 47iT^{2} \)
53 \( 1 + (-0.0579 - 0.0579i)T + 53iT^{2} \)
59 \( 1 - 1.06T + 59T^{2} \)
61 \( 1 - 7.56T + 61T^{2} \)
67 \( 1 + (-5.10 - 5.10i)T + 67iT^{2} \)
71 \( 1 + 15.7iT - 71T^{2} \)
73 \( 1 + (3.90 - 3.90i)T - 73iT^{2} \)
79 \( 1 + 7.85iT - 79T^{2} \)
83 \( 1 + (-8.60 - 8.60i)T + 83iT^{2} \)
89 \( 1 + 5.43T + 89T^{2} \)
97 \( 1 + (2.97 + 2.97i)T + 97iT^{2} \)
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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

−9.771191464628539310581471326241, −9.699869356502723609708928297089, −8.425402250074819157016829321296, −7.71832911146913036246793449264, −7.25975072112147679780899100578, −6.40775456713468100518182753468, −4.93400399696062452191720412352, −3.85985650124947771141574086329, −3.06346925676516545208671135059, −1.49590068810356193713962058281, 0.66379273516993810054535373774, 2.34141690144701761100418010010, 3.29909788955409671729487920552, 4.06724991509641363637444457597, 5.25653759279896623321581070755, 6.81644253929081703549596827925, 7.49459987632511831536487757323, 8.570888243892868126838628698626, 8.811858490000517395996988135031, 9.553091685491803842555459449921

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