Properties

Label 2-930-31.9-c1-0-0
Degree $2$
Conductor $930$
Sign $-0.961 - 0.275i$
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  + (−0.309 + 0.951i)2-s + (−0.978 − 0.207i)3-s + (−0.809 − 0.587i)4-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)6-s + (−0.897 + 0.399i)7-s + (0.809 − 0.587i)8-s + (0.913 + 0.406i)9-s + (0.978 − 0.207i)10-s + (0.250 − 2.38i)11-s + (0.669 + 0.743i)12-s + (−0.151 + 0.168i)13-s + (−0.102 − 0.976i)14-s + (0.309 + 0.951i)15-s + (0.309 + 0.951i)16-s + (0.443 + 4.21i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (−0.564 − 0.120i)3-s + (−0.404 − 0.293i)4-s + (−0.223 − 0.387i)5-s + (0.204 − 0.353i)6-s + (−0.339 + 0.151i)7-s + (0.286 − 0.207i)8-s + (0.304 + 0.135i)9-s + (0.309 − 0.0657i)10-s + (0.0754 − 0.717i)11-s + (0.193 + 0.214i)12-s + (−0.0420 + 0.0467i)13-s + (−0.0274 − 0.261i)14-s + (0.0797 + 0.245i)15-s + (0.0772 + 0.237i)16-s + (0.107 + 1.02i)17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.961 - 0.275i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (691, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.961 - 0.275i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0473060 + 0.336347i\)
\(L(\frac12)\) \(\approx\) \(0.0473060 + 0.336347i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 - 0.951i)T \)
3 \( 1 + (0.978 + 0.207i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (-5.35 - 1.52i)T \)
good7 \( 1 + (0.897 - 0.399i)T + (4.68 - 5.20i)T^{2} \)
11 \( 1 + (-0.250 + 2.38i)T + (-10.7 - 2.28i)T^{2} \)
13 \( 1 + (0.151 - 0.168i)T + (-1.35 - 12.9i)T^{2} \)
17 \( 1 + (-0.443 - 4.21i)T + (-16.6 + 3.53i)T^{2} \)
19 \( 1 + (1.63 + 1.81i)T + (-1.98 + 18.8i)T^{2} \)
23 \( 1 + (2.11 - 1.54i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.494 - 1.52i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 + (4.86 - 8.42i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (11.6 - 2.46i)T + (37.4 - 16.6i)T^{2} \)
43 \( 1 + (-1.56 - 1.73i)T + (-4.49 + 42.7i)T^{2} \)
47 \( 1 + (0.265 + 0.818i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (10.0 + 4.46i)T + (35.4 + 39.3i)T^{2} \)
59 \( 1 + (6.52 + 1.38i)T + (53.8 + 23.9i)T^{2} \)
61 \( 1 - 3.26T + 61T^{2} \)
67 \( 1 + (-4.50 - 7.80i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.43 + 1.52i)T + (47.5 + 52.7i)T^{2} \)
73 \( 1 + (0.974 - 9.27i)T + (-71.4 - 15.1i)T^{2} \)
79 \( 1 + (-0.770 - 7.33i)T + (-77.2 + 16.4i)T^{2} \)
83 \( 1 + (-1.65 + 0.352i)T + (75.8 - 33.7i)T^{2} \)
89 \( 1 + (10.0 + 7.28i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-10.2 - 7.42i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.32726719842278699831379783429, −9.594350015431149297128777302380, −8.485353136940905981938257777579, −8.128481817022217586705201008268, −6.83532780619185787434723304692, −6.28047179431048278030910642626, −5.37764258657713500761655274628, −4.48467439102017092122469231541, −3.30172193315504539484497654809, −1.42027112666139585942505497094, 0.19227561527600727662373566229, 1.94178903419024536809384449000, 3.20377353099288645293411605362, 4.21895028826830070124096777469, 5.11316874289037024044042384629, 6.31479491946493985386965001320, 7.14753223823253795550805584124, 8.001523027062651684941714365126, 9.128724209280242971995740509769, 9.921701754896641586349102880186

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