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Properties

Label 2-930-93.26-c1-0-29
Degree $2$
Conductor $930$
Sign $0.877 + 0.479i$
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 + (1.20 + 1.24i)3-s − 4-s + (0.866 − 0.5i)5-s + (1.24 − 1.20i)6-s + (2.16 − 3.75i)7-s + i·8-s + (−0.110 + 2.99i)9-s + (−0.5 − 0.866i)10-s + (2.88 + 5.00i)11-s + (−1.20 − 1.24i)12-s + (0.829 − 0.478i)13-s + (−3.75 − 2.16i)14-s + (1.66 + 0.478i)15-s + 16-s + (−0.169 + 0.293i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.694 + 0.719i)3-s − 0.5·4-s + (0.387 − 0.223i)5-s + (0.509 − 0.490i)6-s + (0.819 − 1.42i)7-s + 0.353i·8-s + (−0.0367 + 0.999i)9-s + (−0.158 − 0.273i)10-s + (0.871 + 1.50i)11-s + (−0.347 − 0.359i)12-s + (0.230 − 0.132i)13-s + (−1.00 − 0.579i)14-s + (0.429 + 0.123i)15-s + 0.250·16-s + (−0.0410 + 0.0710i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 + 0.479i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.877 + 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.877 + 0.479i$
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: \((2,\ 930,\ (\ :1/2),\ 0.877 + 0.479i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.23647 - 0.571283i\)
\(L(\frac12)\) \(\approx\) \(2.23647 - 0.571283i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.20 - 1.24i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 + (-4.75 + 2.90i)T \)
good7 \( 1 + (-2.16 + 3.75i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.88 - 5.00i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.829 + 0.478i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.169 - 0.293i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.915 + 1.58i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.13T + 23T^{2} \)
29 \( 1 - 7.24T + 29T^{2} \)
37 \( 1 + (5.82 + 3.36i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.47 + 0.849i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.67 - 2.12i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 2.77iT - 47T^{2} \)
53 \( 1 + (4.01 + 6.95i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.461 - 0.266i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 12.6iT - 61T^{2} \)
67 \( 1 + (-6.77 - 11.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.72 + 3.30i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.51 + 3.18i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.30 + 1.33i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.17 - 7.22i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 0.143T + 89T^{2} \)
97 \( 1 + 13.3T + 97T^{2} \)
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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

−10.07323869538125857993219041804, −9.459775771136232866925054363595, −8.445622283743137704370940794705, −7.72553326418024871747836818137, −6.74465163215934560913533571355, −5.15934842418503429254402036565, −4.27977408704746612859229714651, −3.95361355135173220675350586381, −2.37598153260361050744312213127, −1.35583017605986221532301021582, 1.36736879045975593200729343295, 2.61144102372590300138934423043, 3.70430150216372752034684067234, 5.14760310507461891996289059811, 6.24639471202130333036422397215, 6.36007652769593753718792728364, 7.88050182086304933358033070940, 8.409652299825255131053906691225, 8.910603736202704354370022108065, 9.755636718152734066004961083623

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