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Properties

Label 2-930-155.39-c1-0-19
Degree $2$
Conductor $930$
Sign $0.878 + 0.478i$
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 + (2.27 − 3.12i)7-s + (−0.587 − 0.809i)8-s + (0.809 − 0.587i)9-s + (0.0762 − 2.23i)10-s + (0.577 + 0.419i)11-s + (0.951 + 0.309i)12-s + (5.71 − 1.85i)13-s + (−3.12 + 2.27i)14-s + (1.25 + 1.85i)15-s + (0.309 + 0.951i)16-s + (−1.61 − 2.21i)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 + (0.858 − 1.18i)7-s + (−0.207 − 0.286i)8-s + (0.269 − 0.195i)9-s + (0.0241 − 0.706i)10-s + (0.174 + 0.126i)11-s + (0.274 + 0.0892i)12-s + (1.58 − 0.515i)13-s + (−0.835 + 0.606i)14-s + (0.323 + 0.478i)15-s + (0.0772 + 0.237i)16-s + (−0.391 − 0.538i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 + 0.478i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.878 + 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.878 + 0.478i$
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: \((2,\ 930,\ (\ :1/2),\ 0.878 + 0.478i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65463 - 0.421698i\)
\(L(\frac12)\) \(\approx\) \(1.65463 - 0.421698i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.03 + 2.37i)T \)
good7 \( 1 + (-2.27 + 3.12i)T + (-2.16 - 6.65i)T^{2} \)
11 \( 1 + (-0.577 - 0.419i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-5.71 + 1.85i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.61 + 2.21i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.57 - 4.84i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (3.46 + 4.76i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (-0.426 + 1.31i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 + 5.40iT - 37T^{2} \)
41 \( 1 + (3.08 - 9.49i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (-1.97 - 0.642i)T + (34.7 + 25.2i)T^{2} \)
47 \( 1 + (0.107 - 0.0348i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-5.59 - 7.69i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.63 - 5.02i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + 3.71T + 61T^{2} \)
67 \( 1 + 7.24iT - 67T^{2} \)
71 \( 1 + (-5.18 + 3.77i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.79 + 2.47i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-8.60 + 6.25i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-12.9 - 4.19i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (7.59 + 5.51i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (10.3 - 14.2i)T + (-29.9 - 92.2i)T^{2} \)
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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

−10.22354405952338455471247496033, −9.176497226109110201169990691125, −8.056232285572219260855364389536, −7.84438436929728007622434799443, −6.71477362471758072447567626314, −6.07008151662656308111669864770, −4.30282419866292273046824350981, −3.52026433061182971775920307288, −2.29191024744842168155253079001, −1.14394204185313949337482890522, 1.39981649128590211489806345593, 2.25446807786546807145885123738, 3.85733099498353558639320928323, 4.98491141058493196506874060967, 5.81316269511177758185136584636, 6.73932171239443389095662260662, 8.172522401022997892400326778431, 8.564033889089969607186467402479, 8.965188110813139844189817888777, 9.819575854244889637497471061957

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