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Properties

Label 2-930-31.2-c1-0-2
Degree $2$
Conductor $930$
Sign $0.625 - 0.780i$
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  + (−0.809 − 0.587i)2-s + (0.809 − 0.587i)3-s + (0.309 + 0.951i)4-s − 5-s − 6-s + (−0.482 − 1.48i)7-s + (0.309 − 0.951i)8-s + (0.309 − 0.951i)9-s + (0.809 + 0.587i)10-s + (0.945 + 2.91i)11-s + (0.809 + 0.587i)12-s + (−5.04 + 3.66i)13-s + (−0.482 + 1.48i)14-s + (−0.809 + 0.587i)15-s + (−0.809 + 0.587i)16-s + (−2.20 + 6.78i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (0.467 − 0.339i)3-s + (0.154 + 0.475i)4-s − 0.447·5-s − 0.408·6-s + (−0.182 − 0.561i)7-s + (0.109 − 0.336i)8-s + (0.103 − 0.317i)9-s + (0.255 + 0.185i)10-s + (0.285 + 0.877i)11-s + (0.233 + 0.169i)12-s + (−1.39 + 1.01i)13-s + (−0.129 + 0.397i)14-s + (−0.208 + 0.151i)15-s + (−0.202 + 0.146i)16-s + (−0.534 + 1.64i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.625 - 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.625 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.625 - 0.780i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.625 - 0.780i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.808287 + 0.388217i\)
\(L(\frac12)\) \(\approx\) \(0.808287 + 0.388217i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 + 0.587i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + T \)
31 \( 1 + (-5.34 + 1.54i)T \)
good7 \( 1 + (0.482 + 1.48i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (-0.945 - 2.91i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (5.04 - 3.66i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.20 - 6.78i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-5.71 - 4.15i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-0.457 + 1.40i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (6.07 + 4.41i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 + 0.145T + 37T^{2} \)
41 \( 1 + (-0.510 - 0.371i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-7.05 - 5.12i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (6.97 - 5.06i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (3.67 - 11.3i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (2.94 - 2.13i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 - 11.2T + 61T^{2} \)
67 \( 1 + 6.16T + 67T^{2} \)
71 \( 1 + (1.16 - 3.59i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-4.45 - 13.7i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-1.21 + 3.72i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-2.65 - 1.92i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (1.97 + 6.06i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (4.33 + 13.3i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.822772310026995278182701889268, −9.644069154894842989149064100114, −8.496366629411630446107709946954, −7.60940139275678256727303053905, −7.19937104199630567331445286175, −6.15694162423507830192361721384, −4.46776609254115386399038221475, −3.86458509050702895002229533735, −2.51087247161012923754721904943, −1.45498301245681104301336475969, 0.49276822110186639629844544473, 2.56552704731769546937007851641, 3.30930188209172508605223438999, 4.95813297057546279987142667942, 5.40185750660949903875787712383, 6.86492012645454337053160260125, 7.47700012177748400897570528174, 8.301670574547363619409267695310, 9.322579893648185977772681977548, 9.478355478496098075139124781024

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