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Properties

Label 2-930-15.2-c1-0-29
Degree $2$
Conductor $930$
Sign $0.981 - 0.192i$
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  + (−0.707 + 0.707i)2-s + (−0.731 + 1.56i)3-s − 1.00i·4-s + (−1.45 + 1.69i)5-s + (−0.592 − 1.62i)6-s + (1.76 + 1.76i)7-s + (0.707 + 0.707i)8-s + (−1.92 − 2.29i)9-s + (−0.171 − 2.22i)10-s − 5.31i·11-s + (1.56 + 0.731i)12-s + (3.35 − 3.35i)13-s − 2.49·14-s + (−1.60 − 3.52i)15-s − 1.00·16-s + (2.77 − 2.77i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.422 + 0.906i)3-s − 0.500i·4-s + (−0.650 + 0.759i)5-s + (−0.241 − 0.664i)6-s + (0.666 + 0.666i)7-s + (0.250 + 0.250i)8-s + (−0.642 − 0.765i)9-s + (−0.0542 − 0.705i)10-s − 1.60i·11-s + (0.453 + 0.211i)12-s + (0.931 − 0.931i)13-s − 0.666·14-s + (−0.413 − 0.910i)15-s − 0.250·16-s + (0.672 − 0.672i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.192i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 - 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.981 - 0.192i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (497, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.981 - 0.192i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.819700 + 0.0798288i\)
\(L(\frac12)\) \(\approx\) \(0.819700 + 0.0798288i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (0.731 - 1.56i)T \)
5 \( 1 + (1.45 - 1.69i)T \)
31 \( 1 - T \)
good7 \( 1 + (-1.76 - 1.76i)T + 7iT^{2} \)
11 \( 1 + 5.31iT - 11T^{2} \)
13 \( 1 + (-3.35 + 3.35i)T - 13iT^{2} \)
17 \( 1 + (-2.77 + 2.77i)T - 17iT^{2} \)
19 \( 1 + 1.86iT - 19T^{2} \)
23 \( 1 + (3.75 + 3.75i)T + 23iT^{2} \)
29 \( 1 + 3.62T + 29T^{2} \)
37 \( 1 + (-1.42 - 1.42i)T + 37iT^{2} \)
41 \( 1 - 6.12iT - 41T^{2} \)
43 \( 1 + (-6.90 + 6.90i)T - 43iT^{2} \)
47 \( 1 + (5.33 - 5.33i)T - 47iT^{2} \)
53 \( 1 + (7.93 + 7.93i)T + 53iT^{2} \)
59 \( 1 - 8.98T + 59T^{2} \)
61 \( 1 + 3.54T + 61T^{2} \)
67 \( 1 + (-3.66 - 3.66i)T + 67iT^{2} \)
71 \( 1 + 8.57iT - 71T^{2} \)
73 \( 1 + (-0.0629 + 0.0629i)T - 73iT^{2} \)
79 \( 1 + 13.3iT - 79T^{2} \)
83 \( 1 + (-4.31 - 4.31i)T + 83iT^{2} \)
89 \( 1 + 5.00T + 89T^{2} \)
97 \( 1 + (-7.70 - 7.70i)T + 97iT^{2} \)
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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

−10.20161347814811319796276705709, −9.150529784392081399149255797713, −8.327637874565062075779235890343, −7.904142172367132821383137169302, −6.45705155215700657018252792831, −5.84873301725634763581725852108, −5.03980640657090087473994310855, −3.73446564388252658672119921316, −2.87149332350650285452879814295, −0.55650135623334613518580592978, 1.27181815330562808675423561175, 1.87560539817925638478802100940, 3.83405207536001667175605820156, 4.53435907040852968414793761238, 5.73092887167681130354165211786, 6.97320190008650163731254033298, 7.69368742186066902684120861770, 8.141153912342348982218396770353, 9.177176052240248829265092593244, 10.09573606358499745196109662693

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