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Properties

Label 2-930-93.92-c1-0-25
Degree $2$
Conductor $930$
Sign $0.359 - 0.933i$
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  + i·2-s + 1.73·3-s − 4-s + i·5-s + 1.73i·6-s + 2·7-s − i·8-s + 2.99·9-s − 10-s + 3.46·11-s − 1.73·12-s + 2i·14-s + 1.73i·15-s + 16-s − 6.92·17-s + 2.99i·18-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.00·3-s − 0.5·4-s + 0.447i·5-s + 0.707i·6-s + 0.755·7-s − 0.353i·8-s + 0.999·9-s − 0.316·10-s + 1.04·11-s − 0.500·12-s + 0.534i·14-s + 0.447i·15-s + 0.250·16-s − 1.68·17-s + 0.707i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.359 - 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.359 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.359 - 0.933i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (371, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.359 - 0.933i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.99811 + 1.37194i\)
\(L(\frac12)\) \(\approx\) \(1.99811 + 1.37194i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 - 1.73T \)
5 \( 1 - iT \)
31 \( 1 + (-2 + 5.19i)T \)
good7 \( 1 - 2T + 7T^{2} \)
11 \( 1 - 3.46T + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 6.92T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 3.46T + 23T^{2} \)
29 \( 1 + 3.46T + 29T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 12iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 6iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 8T + 67T^{2} \)
71 \( 1 - 12iT - 71T^{2} \)
73 \( 1 + 10.3iT - 73T^{2} \)
79 \( 1 + 10.3iT - 79T^{2} \)
83 \( 1 - 3.46T + 83T^{2} \)
89 \( 1 + 13.8T + 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−9.878517212166056939121410691383, −9.191449693677741650265021358066, −8.544352747672932741786911726157, −7.66480432543471985320740719798, −6.98782957161677007019585848194, −6.17187624400057801178681730923, −4.77025359297147510457006857779, −4.07969902159654154151624307789, −2.88875252916177311357578212890, −1.54619891207124180839820892455, 1.28089911533307140174339136891, 2.22278102258809840334611151867, 3.49262499964166588777279568517, 4.35090861089360639857948319883, 5.14442900275014943979958032356, 6.67911727266497340990275019237, 7.56052931265132680935937363022, 8.662909877130109931397427396597, 8.945878359132911077379612762381, 9.728767105481442917441170858677

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