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Properties

Label 2-930-93.26-c1-0-9
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: $\chi_{930} (491, \cdot )$
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

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

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