Properties

Label 2-930-31.5-c1-0-15
Degree $2$
Conductor $930$
Sign $0.275 + 0.961i$
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  + 2-s + (−0.5 + 0.866i)3-s + 4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (0.5 − 0.866i)7-s + 8-s + (−0.499 − 0.866i)9-s + (−0.5 − 0.866i)10-s + (−2.5 − 4.33i)11-s + (−0.5 + 0.866i)12-s + (−2 − 3.46i)13-s + (0.5 − 0.866i)14-s + 0.999·15-s + 16-s + (1 − 1.73i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.288 + 0.499i)3-s + 0.5·4-s + (−0.223 − 0.387i)5-s + (−0.204 + 0.353i)6-s + (0.188 − 0.327i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.158 − 0.273i)10-s + (−0.753 − 1.30i)11-s + (−0.144 + 0.249i)12-s + (−0.554 − 0.960i)13-s + (0.133 − 0.231i)14-s + 0.258·15-s + 0.250·16-s + (0.242 − 0.420i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.275 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.275 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.275 + 0.961i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (811, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.275 + 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38794 - 1.04654i\)
\(L(\frac12)\) \(\approx\) \(1.38794 - 1.04654i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (-3.5 - 4.33i)T \)
good7 \( 1 + (-0.5 + 0.866i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.5 + 4.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2 + 3.46i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
37 \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2 + 3.46i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 2T + 47T^{2} \)
53 \( 1 + (1.5 + 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4 + 6.92i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.5 + 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 18T + 89T^{2} \)
97 \( 1 + T + 97T^{2} \)
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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

−10.27435505761476451009128576905, −9.040606803200149543593425769758, −8.107670220484163438870640559764, −7.40414112668966146621646301903, −6.12297997229029298466998079656, −5.37593847345605006537868557699, −4.70482354311543036645031617494, −3.58583723919905213035612411632, −2.70545766964029665005559027530, −0.65681548417145443507407451125, 1.83497885094960504484096930921, 2.68985758998949375222145606903, 4.15903527634336895777912034697, 4.91328561041160248966696011618, 5.94985397647227030359309795216, 6.77213598447190992149365961721, 7.55644136112322467177848497916, 8.223942431847154701535969405585, 9.679312627270711473262758799626, 10.26766778397209428236956838925

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