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Properties

Label 2-930-155.149-c1-0-27
Degree $2$
Conductor $930$
Sign $0.107 + 0.994i$
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 + (1.02 − 1.98i)5-s + (0.5 + 0.866i)6-s + (−3.96 + 2.28i)7-s − i·8-s + (0.499 − 0.866i)9-s + (1.98 + 1.02i)10-s + (1.12 − 1.94i)11-s + (−0.866 + 0.5i)12-s + (−4.52 − 2.61i)13-s + (−2.28 − 3.96i)14-s + (−0.107 − 2.23i)15-s + 16-s + (1.87 − 1.08i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.499 − 0.288i)3-s − 0.5·4-s + (0.457 − 0.889i)5-s + (0.204 + 0.353i)6-s + (−1.49 + 0.865i)7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (0.628 + 0.323i)10-s + (0.338 − 0.586i)11-s + (−0.249 + 0.144i)12-s + (−1.25 − 0.724i)13-s + (−0.611 − 1.05i)14-s + (−0.0278 − 0.576i)15-s + 0.250·16-s + (0.455 − 0.262i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.107 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.107 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.107 + 0.994i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.107 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.784402 - 0.703979i\)
\(L(\frac12)\) \(\approx\) \(0.784402 - 0.703979i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.02 + 1.98i)T \)
31 \( 1 + (-1.86 + 5.24i)T \)
good7 \( 1 + (3.96 - 2.28i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.12 + 1.94i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.52 + 2.61i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.87 + 1.08i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.884 - 1.53i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 7.60iT - 23T^{2} \)
29 \( 1 + 3.56T + 29T^{2} \)
37 \( 1 + (2.84 - 1.64i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.66 + 2.87i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.967 - 0.558i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 1.34iT - 47T^{2} \)
53 \( 1 + (4.54 + 2.62i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.77 + 4.80i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 5.98T + 61T^{2} \)
67 \( 1 + (-7.84 - 4.52i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.71 - 9.90i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (13.9 + 8.06i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.15 + 2.00i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-15.1 - 8.74i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 0.683T + 89T^{2} \)
97 \( 1 + 7.66iT - 97T^{2} \)
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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

−9.649288056144897041975451235653, −8.979456507232300153219146920588, −8.302354482299229971140249752476, −7.35394492572376022413929451268, −6.28171179168741880798272790887, −5.77556077442104268014287215081, −4.76038403365524361920936982272, −3.39702971370237190075581886259, −2.40318665790065721402823357482, −0.44432653411737126309898552280, 1.81065787142395281886077755634, 3.03644894136309226005267363321, 3.58533458139964130307690747458, 4.68406458093191148039997761445, 6.01287797686759406683243891019, 7.11585854747058623459861059697, 7.43505343157053895561906709566, 9.142952793425271300978250678374, 9.663865308201264461893905016067, 10.04547655396759844433184603227

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