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Properties

Label 2-930-155.4-c1-0-20
Degree $2$
Conductor $930$
Sign $0.588 + 0.808i$
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 + (0.618 − 2.14i)5-s − 0.999·6-s + (1.53 + 2.11i)7-s + (0.587 − 0.809i)8-s + (0.809 + 0.587i)9-s + (−0.0762 − 2.23i)10-s + (−1.18 + 0.858i)11-s + (−0.951 + 0.309i)12-s + (4.52 + 1.47i)13-s + (2.11 + 1.53i)14-s + (−1.25 + 1.85i)15-s + (0.309 − 0.951i)16-s + (2.98 − 4.10i)17-s + ⋯
L(s)  = 1  + (0.672 − 0.218i)2-s + (−0.549 − 0.178i)3-s + (0.404 − 0.293i)4-s + (0.276 − 0.961i)5-s − 0.408·6-s + (0.579 + 0.797i)7-s + (0.207 − 0.286i)8-s + (0.269 + 0.195i)9-s + (−0.0241 − 0.706i)10-s + (−0.356 + 0.258i)11-s + (−0.274 + 0.0892i)12-s + (1.25 + 0.407i)13-s + (0.564 + 0.409i)14-s + (−0.323 + 0.478i)15-s + (0.0772 − 0.237i)16-s + (0.723 − 0.996i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.588 + 0.808i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.588 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.588 + 0.808i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.588 + 0.808i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.07203 - 1.05388i\)
\(L(\frac12)\) \(\approx\) \(2.07203 - 1.05388i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.618 + 2.14i)T \)
31 \( 1 + (1.69 - 5.30i)T \)
good7 \( 1 + (-1.53 - 2.11i)T + (-2.16 + 6.65i)T^{2} \)
11 \( 1 + (1.18 - 0.858i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-4.52 - 1.47i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-2.98 + 4.10i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.574 - 1.76i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-2.59 + 3.57i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (-0.334 - 1.02i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 + 4.67iT - 37T^{2} \)
41 \( 1 + (2.57 + 7.91i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (-4.48 + 1.45i)T + (34.7 - 25.2i)T^{2} \)
47 \( 1 + (9.18 + 2.98i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.866 - 1.19i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.33 - 4.11i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + 0.894T + 61T^{2} \)
67 \( 1 + 2.19iT - 67T^{2} \)
71 \( 1 + (-9.22 - 6.70i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-7.37 - 10.1i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (2.95 + 2.14i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-1.39 + 0.453i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (1.40 - 1.02i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (1.18 + 1.63i)T + (-29.9 + 92.2i)T^{2} \)
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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

−10.07811940513300604467727229604, −9.040108567441353313411894597013, −8.390222836750277677409100633383, −7.26601065481584680407981806304, −6.19662885218350013278419157250, −5.33179799295808365009528978090, −4.96132921087637841894701443303, −3.73716465553624905746223499866, −2.24314401549574483827835518363, −1.14826872396199274015345944753, 1.44009957559845878758583340623, 3.12676528652730752322854133076, 3.86992164006762750527967422017, 5.00359449011917344262128688886, 5.95942584597906931099643692193, 6.49516499048032907292223597575, 7.59688987556448767085245143398, 8.144293232073177963564187410240, 9.628441732221058915008900843462, 10.49082688241648129219941452905

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