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Properties

Label 2-930-15.2-c1-0-26
Degree $2$
Conductor $930$
Sign $0.875 + 0.483i$
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 + (−1.56 + 0.731i)3-s − 1.00i·4-s + (1.45 − 1.69i)5-s + (−0.592 + 1.62i)6-s + (1.76 + 1.76i)7-s + (−0.707 − 0.707i)8-s + (1.92 − 2.29i)9-s + (−0.171 − 2.22i)10-s + 5.31i·11-s + (0.731 + 1.56i)12-s + (3.35 − 3.35i)13-s + 2.49·14-s + (−1.04 + 3.73i)15-s − 1.00·16-s + (−2.77 + 2.77i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.906 + 0.422i)3-s − 0.500i·4-s + (0.650 − 0.759i)5-s + (−0.241 + 0.664i)6-s + (0.666 + 0.666i)7-s + (−0.250 − 0.250i)8-s + (0.642 − 0.765i)9-s + (−0.0542 − 0.705i)10-s + 1.60i·11-s + (0.211 + 0.453i)12-s + (0.931 − 0.931i)13-s + 0.666·14-s + (−0.269 + 0.963i)15-s − 0.250·16-s + (−0.672 + 0.672i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 + 0.483i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.875 + 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.875 + 0.483i$
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: \((2,\ 930,\ (\ :1/2),\ 0.875 + 0.483i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86749 - 0.481033i\)
\(L(\frac12)\) \(\approx\) \(1.86749 - 0.481033i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.56 - 0.731i)T \)
5 \( 1 + (-1.45 + 1.69i)T \)
31 \( 1 - T \)
good7 \( 1 + (-1.76 - 1.76i)T + 7iT^{2} \)
11 \( 1 - 5.31iT - 11T^{2} \)
13 \( 1 + (-3.35 + 3.35i)T - 13iT^{2} \)
17 \( 1 + (2.77 - 2.77i)T - 17iT^{2} \)
19 \( 1 + 1.86iT - 19T^{2} \)
23 \( 1 + (-3.75 - 3.75i)T + 23iT^{2} \)
29 \( 1 - 3.62T + 29T^{2} \)
37 \( 1 + (-1.42 - 1.42i)T + 37iT^{2} \)
41 \( 1 + 6.12iT - 41T^{2} \)
43 \( 1 + (-6.90 + 6.90i)T - 43iT^{2} \)
47 \( 1 + (-5.33 + 5.33i)T - 47iT^{2} \)
53 \( 1 + (-7.93 - 7.93i)T + 53iT^{2} \)
59 \( 1 + 8.98T + 59T^{2} \)
61 \( 1 + 3.54T + 61T^{2} \)
67 \( 1 + (-3.66 - 3.66i)T + 67iT^{2} \)
71 \( 1 - 8.57iT - 71T^{2} \)
73 \( 1 + (-0.0629 + 0.0629i)T - 73iT^{2} \)
79 \( 1 + 13.3iT - 79T^{2} \)
83 \( 1 + (4.31 + 4.31i)T + 83iT^{2} \)
89 \( 1 - 5.00T + 89T^{2} \)
97 \( 1 + (-7.70 - 7.70i)T + 97iT^{2} \)
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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

−10.30912318084809344780364434361, −9.262248553470778129105271594202, −8.695903208211495733519875373707, −7.27951790209779082902684994513, −6.14719700364086145374299169362, −5.42448255478317070135368388495, −4.81715576532043727528495025326, −3.98530224266395506909024078356, −2.28969164499006413269023151615, −1.19005303346506620781210641321, 1.16625158220418819067293424885, 2.72682082929841428973143497452, 4.07763847594041890796073569573, 5.01225505466745637748758920234, 6.13904696466156332078410866517, 6.39535101592095776777117932902, 7.31892915212053731052560320392, 8.228812987838903878268341228717, 9.211341612386207885686434124905, 10.52768122285223250219141791542

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