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Properties

Label 2-930-155.109-c1-0-2
Degree $2$
Conductor $930$
Sign $-0.0229 - 0.999i$
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  + (−0.587 − 0.809i)2-s + (0.587 − 0.809i)3-s + (−0.309 + 0.951i)4-s + 2.23·5-s − 6-s + (−3.07 − i)7-s + (0.951 − 0.309i)8-s + (−0.309 − 0.951i)9-s + (−1.31 − 1.80i)10-s + (−1.42 + 4.39i)11-s + (0.587 + 0.809i)12-s + (−4.02 + 5.54i)13-s + (1 + 3.07i)14-s + (1.31 − 1.80i)15-s + (−0.809 − 0.587i)16-s + (−4.61 + 1.5i)17-s + ⋯
L(s)  = 1  + (−0.415 − 0.572i)2-s + (0.339 − 0.467i)3-s + (−0.154 + 0.475i)4-s + 0.999·5-s − 0.408·6-s + (−1.16 − 0.377i)7-s + (0.336 − 0.109i)8-s + (−0.103 − 0.317i)9-s + (−0.415 − 0.572i)10-s + (−0.430 + 1.32i)11-s + (0.169 + 0.233i)12-s + (−1.11 + 1.53i)13-s + (0.267 + 0.822i)14-s + (0.339 − 0.467i)15-s + (−0.202 − 0.146i)16-s + (−1.11 + 0.363i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0229 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0229 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.0229 - 0.999i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.0229 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.348784 + 0.356870i\)
\(L(\frac12)\) \(\approx\) \(0.348784 + 0.356870i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.587 + 0.809i)T \)
3 \( 1 + (-0.587 + 0.809i)T \)
5 \( 1 - 2.23T \)
31 \( 1 + (3.30 + 4.47i)T \)
good7 \( 1 + (3.07 + i)T + (5.66 + 4.11i)T^{2} \)
11 \( 1 + (1.42 - 4.39i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (4.02 - 5.54i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (4.61 - 1.5i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (3.61 - 2.62i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (0.812 - 0.263i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (-0.118 + 0.0857i)T + (8.96 - 27.5i)T^{2} \)
37 \( 1 + 4.14iT - 37T^{2} \)
41 \( 1 + (4.61 - 3.35i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + (-2.35 - 3.23i)T + (-13.2 + 40.8i)T^{2} \)
47 \( 1 + (-4.75 + 6.54i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (5.42 - 1.76i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (7.97 + 5.79i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 - 7.23T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 + (1.23 + 3.80i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-14.2 - 4.61i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-1.57 - 4.84i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (4.25 + 5.85i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (4.85 - 14.9i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-16.1 - 5.23i)T + (78.4 + 57.0i)T^{2} \)
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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

−10.02009697142836362920265700255, −9.523013497982570099931969913348, −9.005445086394599624277233222975, −7.72017553389055296982476747949, −6.82032032928795080146084395312, −6.39223177412606570268791475887, −4.81940049270113258530077191456, −3.85181236363396622659033056207, −2.29857787025764382144043632028, −1.99638049850901129754191700999, 0.22889774422638586514525132731, 2.47862404844732387518748661996, 3.15754825578087716076852155638, 4.84859859301969935672611393652, 5.64370727366593582958689665468, 6.33210014885245132440833239389, 7.27514201438354948534752570350, 8.494628306753477909359532713901, 8.955614624956895822281273610152, 9.783330969998762851221834831598

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