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Properties

Label 2-930-155.129-c1-0-18
Degree $2$
Conductor $930$
Sign $-0.144 + 0.989i$
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 + (−1.86 − 1.23i)5-s + (−0.5 + 0.866i)6-s + (2.29 + 1.32i)7-s + i·8-s + (0.499 + 0.866i)9-s + (−1.23 + 1.86i)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 + 2i)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 + (−0.834 − 0.550i)5-s + (−0.204 + 0.353i)6-s + (0.866 + 0.499i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.389 + 0.590i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.144 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.144 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.144 + 0.989i$
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: \((2,\ 930,\ (\ :1/2),\ -0.144 + 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.799371 - 0.924334i\)
\(L(\frac12)\) \(\approx\) \(0.799371 - 0.924334i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.86 + 1.23i)T \)
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

−10.17838208281642467650488374745, −8.740395839681200816195798315678, −8.309078201119339946587910933899, −7.64049415384359349806168095704, −6.12143747885621098061980427092, −5.42261680191107320191236396021, −4.35752124519406388710076940027, −3.56933351639197654653793080497, −1.95933162340340197412530790143, −0.807700729561835969870676241465, 1.14741143626925617407257154156, 3.35140856199867041525768955024, 4.15463754937275653501116318210, 5.05996338139564570661624414863, 6.02727317213898855172979683389, 7.05067717757045753354076322737, 7.55529433577924046355349083077, 8.497147613376360936116800301092, 9.332011938497915534869899395216, 10.41423115814379103454506890449

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