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Properties

Label 2-930-15.2-c1-0-17
Degree $2$
Conductor $930$
Sign $0.946 - 0.322i$
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 + (−1.49 − 0.877i)3-s − 1.00i·4-s + (1.98 + 1.03i)5-s + (−1.67 + 0.435i)6-s + (2.39 + 2.39i)7-s + (−0.707 − 0.707i)8-s + (1.46 + 2.62i)9-s + (2.13 − 0.673i)10-s − 0.0383i·11-s + (−0.877 + 1.49i)12-s + (−2.42 + 2.42i)13-s + 3.38·14-s + (−2.05 − 3.28i)15-s − 1.00·16-s + (−3.70 + 3.70i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.862 − 0.506i)3-s − 0.500i·4-s + (0.887 + 0.461i)5-s + (−0.684 + 0.177i)6-s + (0.905 + 0.905i)7-s + (−0.250 − 0.250i)8-s + (0.487 + 0.873i)9-s + (0.674 − 0.213i)10-s − 0.0115i·11-s + (−0.253 + 0.431i)12-s + (−0.671 + 0.671i)13-s + 0.905·14-s + (−0.531 − 0.847i)15-s − 0.250·16-s + (−0.898 + 0.898i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.322i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 - 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.946 - 0.322i$
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: \((2,\ 930,\ (\ :1/2),\ 0.946 - 0.322i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.74299 + 0.289001i\)
\(L(\frac12)\) \(\approx\) \(1.74299 + 0.289001i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.49 + 0.877i)T \)
5 \( 1 + (-1.98 - 1.03i)T \)
31 \( 1 + T \)
good7 \( 1 + (-2.39 - 2.39i)T + 7iT^{2} \)
11 \( 1 + 0.0383iT - 11T^{2} \)
13 \( 1 + (2.42 - 2.42i)T - 13iT^{2} \)
17 \( 1 + (3.70 - 3.70i)T - 17iT^{2} \)
19 \( 1 - 7.16iT - 19T^{2} \)
23 \( 1 + (-3.94 - 3.94i)T + 23iT^{2} \)
29 \( 1 + 3.49T + 29T^{2} \)
37 \( 1 + (4.89 + 4.89i)T + 37iT^{2} \)
41 \( 1 + 10.1iT - 41T^{2} \)
43 \( 1 + (6.93 - 6.93i)T - 43iT^{2} \)
47 \( 1 + (-7.35 + 7.35i)T - 47iT^{2} \)
53 \( 1 + (-0.287 - 0.287i)T + 53iT^{2} \)
59 \( 1 - 15.0T + 59T^{2} \)
61 \( 1 - 5.16T + 61T^{2} \)
67 \( 1 + (-5.80 - 5.80i)T + 67iT^{2} \)
71 \( 1 + 9.58iT - 71T^{2} \)
73 \( 1 + (-9.96 + 9.96i)T - 73iT^{2} \)
79 \( 1 + 16.7iT - 79T^{2} \)
83 \( 1 + (-8.18 - 8.18i)T + 83iT^{2} \)
89 \( 1 + 3.51T + 89T^{2} \)
97 \( 1 + (2.47 + 2.47i)T + 97iT^{2} \)
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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

−10.42896481903713534810903592877, −9.474719552889132068288718121148, −8.516304024057011265087184056287, −7.31320534633295648928714290143, −6.47494146546337172023910702455, −5.52930798747843501105939833995, −5.21053175930480633010019988968, −3.85238477513523727683087344106, −2.08416578846940028485695335056, −1.81538645896648980782631328209, 0.796147995346928768717227471685, 2.60709285292775945109360044187, 4.21274282768030558823491959342, 5.05638681976378718621529910386, 5.20917000726812470325592860594, 6.70361742422508687891429410048, 7.03863544895424315725754482986, 8.358976526823467654840492739505, 9.257569253214731953361457465292, 10.06396301353787752283170436091

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