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Properties

Label 2-930-155.129-c1-0-1
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  + i·2-s + (0.866 + 0.5i)3-s − 4-s + (−0.133 − 2.23i)5-s + (−0.5 + 0.866i)6-s + (−2.29 − 1.32i)7-s − i·8-s + (0.499 + 0.866i)9-s + (2.23 − 0.133i)10-s + (0.322 + 0.559i)11-s + (−0.866 − 0.5i)12-s + (−4.58 + 2.64i)13-s + (1.32 − 2.29i)14-s + (1 − 1.99i)15-s + 16-s + (−5.19 − 3i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.499 + 0.288i)3-s − 0.5·4-s + (−0.0599 − 0.998i)5-s + (−0.204 + 0.353i)6-s + (−0.866 − 0.499i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (0.705 − 0.0423i)10-s + (0.0973 + 0.168i)11-s + (−0.249 − 0.144i)12-s + (−1.27 + 0.733i)13-s + (0.353 − 0.612i)14-s + (0.258 − 0.516i)15-s + 0.250·16-s + (−1.26 − 0.727i)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: $\chi_{930} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.878 + 0.478i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0287580 - 0.112917i\)
\(L(\frac12)\) \(\approx\) \(0.0287580 - 0.112917i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (0.133 + 2.23i)T \)
31 \( 1 + (-1.14 + 5.44i)T \)
good7 \( 1 + (2.29 + 1.32i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.322 - 0.559i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.58 - 2.64i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (5.19 + 3i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.64 - 4.58i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 9.29iT - 23T^{2} \)
29 \( 1 + 2.64T + 29T^{2} \)
37 \( 1 + (-3.46 - 2i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.64 + 2.85i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (9.77 + 5.64i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 9.29iT - 47T^{2} \)
53 \( 1 + (1.98 - 1.14i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.322 - 0.559i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 + (-9.77 + 5.64i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.29 + 5.70i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.70 - 3.29i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.29 + 12.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.29 + 4.79i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 3.29T + 89T^{2} \)
97 \( 1 + 11.2iT - 97T^{2} \)
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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

−10.03503310206099725098515730716, −9.523010065865365026625990977236, −9.035048880395459558699295253456, −7.945926762119248794505007553175, −7.28478262107651095123556052515, −6.37411389901823225394343026361, −5.21984090439864470553426168536, −4.40869330381569450173305628594, −3.61719302778134991725930632627, −1.97704610034510650731432036118, 0.04702076176915943560845140555, 2.38289266022585156083187963647, 2.70057051186304567874734785066, 3.83793470666451503514538954768, 5.00364831543943178488478151674, 6.49159712575180428243645294769, 6.78469988314511997469536412088, 8.116962443047445634343157406620, 8.808713228026502463421316791627, 9.743414331016666895341463411354

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