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Properties

Label 2-930-15.8-c1-0-13
Degree $2$
Conductor $930$
Sign $0.992 + 0.122i$
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 + (0.853 − 1.50i)3-s + 1.00i·4-s + (−2.20 + 0.358i)5-s + (−1.66 + 0.461i)6-s + (−1.72 + 1.72i)7-s + (0.707 − 0.707i)8-s + (−1.54 − 2.57i)9-s + (1.81 + 1.30i)10-s + 0.339i·11-s + (1.50 + 0.853i)12-s + (2.77 + 2.77i)13-s + 2.43·14-s + (−1.34 + 3.63i)15-s − 1.00·16-s + (−0.287 − 0.287i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.492 − 0.870i)3-s + 0.500i·4-s + (−0.987 + 0.160i)5-s + (−0.681 + 0.188i)6-s + (−0.650 + 0.650i)7-s + (0.250 − 0.250i)8-s + (−0.513 − 0.857i)9-s + (0.573 + 0.413i)10-s + 0.102i·11-s + (0.435 + 0.246i)12-s + (0.769 + 0.769i)13-s + 0.650·14-s + (−0.346 + 0.937i)15-s − 0.250·16-s + (−0.0697 − 0.0697i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.122i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.992 + 0.122i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (683, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.992 + 0.122i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.992464 - 0.0608655i\)
\(L(\frac12)\) \(\approx\) \(0.992464 - 0.0608655i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.853 + 1.50i)T \)
5 \( 1 + (2.20 - 0.358i)T \)
31 \( 1 - T \)
good7 \( 1 + (1.72 - 1.72i)T - 7iT^{2} \)
11 \( 1 - 0.339iT - 11T^{2} \)
13 \( 1 + (-2.77 - 2.77i)T + 13iT^{2} \)
17 \( 1 + (0.287 + 0.287i)T + 17iT^{2} \)
19 \( 1 - 5.47iT - 19T^{2} \)
23 \( 1 + (-3.99 + 3.99i)T - 23iT^{2} \)
29 \( 1 - 6.27T + 29T^{2} \)
37 \( 1 + (-0.324 + 0.324i)T - 37iT^{2} \)
41 \( 1 - 8.78iT - 41T^{2} \)
43 \( 1 + (-6.31 - 6.31i)T + 43iT^{2} \)
47 \( 1 + (-2.27 - 2.27i)T + 47iT^{2} \)
53 \( 1 + (-7.53 + 7.53i)T - 53iT^{2} \)
59 \( 1 - 0.628T + 59T^{2} \)
61 \( 1 + 11.4T + 61T^{2} \)
67 \( 1 + (-0.339 + 0.339i)T - 67iT^{2} \)
71 \( 1 + 1.05iT - 71T^{2} \)
73 \( 1 + (-6.50 - 6.50i)T + 73iT^{2} \)
79 \( 1 + 3.87iT - 79T^{2} \)
83 \( 1 + (6.20 - 6.20i)T - 83iT^{2} \)
89 \( 1 + 13.1T + 89T^{2} \)
97 \( 1 + (-0.395 + 0.395i)T - 97iT^{2} \)
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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

−9.941316410501796141429391308866, −9.000681388247059095716660276794, −8.451715427495517139427226029470, −7.73663767886073946287099343132, −6.75438616006329541185336256757, −6.13591015591222430425811669339, −4.38306915744071993495330764250, −3.34929386111565874683420061583, −2.56294877904484499492571047123, −1.11709328106142984636917366412, 0.64269396492481409334658078981, 2.91486562162842413963033552953, 3.77749689111856976007340224318, 4.69776788225066973656391675068, 5.71153632338425824997836439011, 7.00041150478912991374592743958, 7.58667180138597573939149837737, 8.658143193801248315193756010719, 8.951891308485641606001114033015, 10.06775804054121580396331992911

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