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Properties

Label 2-930-155.39-c1-0-23
Degree $2$
Conductor $930$
Sign $0.980 - 0.197i$
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  + (0.951 + 0.309i)2-s + (0.951 − 0.309i)3-s + (0.809 + 0.587i)4-s + (2.21 + 0.295i)5-s + 0.999·6-s + (1.19 − 1.64i)7-s + (0.587 + 0.809i)8-s + (0.809 − 0.587i)9-s + (2.01 + 0.966i)10-s + (1.28 + 0.930i)11-s + (0.951 + 0.309i)12-s + (−3.65 + 1.18i)13-s + (1.64 − 1.19i)14-s + (2.19 − 0.403i)15-s + (0.309 + 0.951i)16-s + (2.28 + 3.13i)17-s + ⋯
L(s)  = 1  + (0.672 + 0.218i)2-s + (0.549 − 0.178i)3-s + (0.404 + 0.293i)4-s + (0.991 + 0.132i)5-s + 0.408·6-s + (0.453 − 0.623i)7-s + (0.207 + 0.286i)8-s + (0.269 − 0.195i)9-s + (0.637 + 0.305i)10-s + (0.385 + 0.280i)11-s + (0.274 + 0.0892i)12-s + (−1.01 + 0.329i)13-s + (0.440 − 0.320i)14-s + (0.567 − 0.104i)15-s + (0.0772 + 0.237i)16-s + (0.553 + 0.761i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.197i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.980 - 0.197i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.980 - 0.197i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.40680 + 0.340545i\)
\(L(\frac12)\) \(\approx\) \(3.40680 + 0.340545i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.951 - 0.309i)T \)
3 \( 1 + (-0.951 + 0.309i)T \)
5 \( 1 + (-2.21 - 0.295i)T \)
31 \( 1 + (5.56 + 0.110i)T \)
good7 \( 1 + (-1.19 + 1.64i)T + (-2.16 - 6.65i)T^{2} \)
11 \( 1 + (-1.28 - 0.930i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (3.65 - 1.18i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-2.28 - 3.13i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.925 - 2.84i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (4.65 + 6.40i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (-2.39 + 7.37i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 + 5.90iT - 37T^{2} \)
41 \( 1 + (1.05 - 3.26i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (-4.75 - 1.54i)T + (34.7 + 25.2i)T^{2} \)
47 \( 1 + (11.4 - 3.73i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (2.25 + 3.10i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.79 - 11.6i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + 4.76T + 61T^{2} \)
67 \( 1 - 3.26iT - 67T^{2} \)
71 \( 1 + (-5.71 + 4.14i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.829 + 1.14i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-9.10 + 6.61i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (12.4 + 4.04i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (-10.7 - 7.82i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-4.12 + 5.67i)T + (-29.9 - 92.2i)T^{2} \)
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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

−10.09164282868302651806447467468, −9.360467861386204704291164688063, −8.196746519490857414129409417362, −7.53506638006062566509724801266, −6.55839524228750592405726832284, −5.87997862420239128590784067348, −4.66719614173271087195589435050, −3.92657895875796885048465663229, −2.54286650747835821572230691151, −1.67683016696528945069547040693, 1.62988761981719583754626927867, 2.58105379531698920306659152125, 3.54121870006241757306223798332, 5.07180481753840284406093642200, 5.26634111384543590702440462714, 6.49630761033945283536660369539, 7.43838070000542199964993845985, 8.496983172468642488920000518090, 9.413451389875906317558562641513, 9.876177048624404865553470099926

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