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Properties

Label 2-930-155.37-c1-0-31
Degree $2$
Conductor $930$
Sign $-0.746 + 0.665i$
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 + (0.258 − 0.965i)3-s + 1.00i·4-s + (−1.66 + 1.49i)5-s + (0.866 − 0.500i)6-s + (1.33 − 4.97i)7-s + (−0.707 + 0.707i)8-s + (−0.866 − 0.499i)9-s + (−2.23 − 0.117i)10-s + (−3.19 − 1.84i)11-s + (0.965 + 0.258i)12-s + (−5.79 + 1.55i)13-s + (4.46 − 2.57i)14-s + (1.01 + 1.99i)15-s − 1.00·16-s + (−1.42 − 0.381i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.149 − 0.557i)3-s + 0.500i·4-s + (−0.743 + 0.669i)5-s + (0.353 − 0.204i)6-s + (0.504 − 1.88i)7-s + (−0.250 + 0.250i)8-s + (−0.288 − 0.166i)9-s + (−0.706 − 0.0370i)10-s + (−0.964 − 0.556i)11-s + (0.278 + 0.0747i)12-s + (−1.60 + 0.430i)13-s + (1.19 − 0.688i)14-s + (0.262 + 0.514i)15-s − 0.250·16-s + (−0.345 − 0.0925i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.746 + 0.665i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.746 + 0.665i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.189928 - 0.498640i\)
\(L(\frac12)\) \(\approx\) \(0.189928 - 0.498640i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 + (1.66 - 1.49i)T \)
31 \( 1 + (5.35 + 1.53i)T \)
good7 \( 1 + (-1.33 + 4.97i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (3.19 + 1.84i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (5.79 - 1.55i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (1.42 + 0.381i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (6.55 - 3.78i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.00 - 4.00i)T + 23iT^{2} \)
29 \( 1 - 3.10T + 29T^{2} \)
37 \( 1 + (4.73 + 1.26i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-5.81 + 10.0i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.29 - 4.82i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (1.03 + 1.03i)T + 47iT^{2} \)
53 \( 1 + (-6.85 + 1.83i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-5.40 + 3.11i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 8.33iT - 61T^{2} \)
67 \( 1 + (-7.51 + 2.01i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (2.17 - 3.76i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-9.85 + 2.64i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-1.61 - 2.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.94 - 2.39i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + 11.3T + 89T^{2} \)
97 \( 1 + (4.56 + 4.56i)T + 97iT^{2} \)
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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

−9.922114119784066856906538022969, −8.429692076045677308466483396155, −7.75404263222836034035452822016, −7.17837208096718277037517297979, −6.72671025518823931913806033836, −5.29078573150746507575045123463, −4.29988086095389703582209711438, −3.55138675309289741787620494468, −2.25396628022848328762104783205, −0.18959993689788149395246041341, 2.28819286432695940544771704547, 2.78021238299370018710148275121, 4.46780129979244854496867698240, 4.93386449841781419267657626715, 5.53914091447737538334827414508, 6.99544951559391665348363563589, 8.215389405192021891309480932637, 8.750495703187229510125212920808, 9.504750413586550912434366353095, 10.52043167096851284933240892717

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