Properties

Label 2-930-93.68-c1-0-31
Degree $2$
Conductor $930$
Sign $-0.0299 + 0.999i$
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  i·2-s + (−0.556 + 1.64i)3-s − 4-s + (0.866 + 0.5i)5-s + (1.64 + 0.556i)6-s + (−1.24 − 2.15i)7-s + i·8-s + (−2.38 − 1.82i)9-s + (0.5 − 0.866i)10-s + (−0.404 + 0.700i)11-s + (0.556 − 1.64i)12-s + (3.21 + 1.85i)13-s + (−2.15 + 1.24i)14-s + (−1.30 + 1.14i)15-s + 16-s + (−3.37 − 5.84i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.321 + 0.947i)3-s − 0.5·4-s + (0.387 + 0.223i)5-s + (0.669 + 0.227i)6-s + (−0.471 − 0.816i)7-s + 0.353i·8-s + (−0.793 − 0.608i)9-s + (0.158 − 0.273i)10-s + (−0.121 + 0.211i)11-s + (0.160 − 0.473i)12-s + (0.892 + 0.515i)13-s + (−0.577 + 0.333i)14-s + (−0.336 + 0.294i)15-s + 0.250·16-s + (−0.818 − 1.41i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0299 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0299 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.0299 + 0.999i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.0299 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.714380 - 0.736074i\)
\(L(\frac12)\) \(\approx\) \(0.714380 - 0.736074i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (0.556 - 1.64i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
31 \( 1 + (5.56 - 0.290i)T \)
good7 \( 1 + (1.24 + 2.15i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.404 - 0.700i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.21 - 1.85i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.37 + 5.84i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.45 + 5.98i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 2.81T + 23T^{2} \)
29 \( 1 - 9.79T + 29T^{2} \)
37 \( 1 + (3.94 - 2.27i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.77 - 1.60i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.26 + 3.04i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 12.3iT - 47T^{2} \)
53 \( 1 + (-4.33 + 7.50i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.80 + 1.04i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 1.93iT - 61T^{2} \)
67 \( 1 + (-1.82 + 3.16i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.69 + 3.28i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (11.0 + 6.39i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.86 - 2.81i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.60 + 11.4i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 7.63T + 89T^{2} \)
97 \( 1 - 3.29T + 97T^{2} \)
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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

−10.11870574311270589893164538400, −9.048225327904330645023450514911, −8.814713360771624817537816637627, −7.05558123143114420711864314174, −6.46792613973433924978949114051, −5.12373658985744866178455085538, −4.44584891608066377331912122745, −3.47904138265220335570256883154, −2.46631844169467737586241578037, −0.53507982917469891793000639278, 1.38270026233505777762356708155, 2.72941423720126118192560433838, 4.16811639434823149721049815678, 5.60790325909080022047454915294, 6.00530925512076245195820990213, 6.60306119465562912526981495154, 7.81792306579423769713463118302, 8.542556176424792420336541045783, 9.000985373765238150648870375455, 10.39500100304200437407310304564

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