Properties

Label 2-950-25.21-c1-0-7
Degree $2$
Conductor $950$
Sign $0.477 - 0.878i$
Analytic cond. $7.58578$
Root an. cond. $2.75423$
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  + (−0.309 + 0.951i)2-s + (−2.30 − 1.67i)3-s + (−0.809 − 0.587i)4-s + (−0.482 + 2.18i)5-s + (2.30 − 1.67i)6-s − 2.79·7-s + (0.809 − 0.587i)8-s + (1.58 + 4.86i)9-s + (−1.92 − 1.13i)10-s + (1.37 − 4.23i)11-s + (0.880 + 2.70i)12-s + (−1.87 − 5.76i)13-s + (0.862 − 2.65i)14-s + (4.76 − 4.22i)15-s + (0.309 + 0.951i)16-s + (−5.08 + 3.69i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (−1.33 − 0.966i)3-s + (−0.404 − 0.293i)4-s + (−0.215 + 0.976i)5-s + (0.941 − 0.683i)6-s − 1.05·7-s + (0.286 − 0.207i)8-s + (0.527 + 1.62i)9-s + (−0.609 − 0.358i)10-s + (0.414 − 1.27i)11-s + (0.254 + 0.782i)12-s + (−0.519 − 1.59i)13-s + (0.230 − 0.709i)14-s + (1.23 − 1.09i)15-s + (0.0772 + 0.237i)16-s + (−1.23 + 0.896i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 - 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.477 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.477 - 0.878i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.477 - 0.878i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.356492 + 0.211950i\)
\(L(\frac12)\) \(\approx\) \(0.356492 + 0.211950i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (0.482 - 2.18i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
good3 \( 1 + (2.30 + 1.67i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + 2.79T + 7T^{2} \)
11 \( 1 + (-1.37 + 4.23i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (1.87 + 5.76i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (5.08 - 3.69i)T + (5.25 - 16.1i)T^{2} \)
23 \( 1 + (1.98 - 6.11i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (1.96 + 1.42i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-4.98 + 3.61i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.176 - 0.542i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.23 - 3.80i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 11.1T + 43T^{2} \)
47 \( 1 + (-5.08 - 3.69i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-6.48 - 4.70i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.88 - 8.86i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (1.56 - 4.82i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-7.82 + 5.68i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (0.755 + 0.548i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.75 - 5.39i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-3.18 - 2.31i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-1.42 + 1.03i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (0.697 - 2.14i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-7.35 - 5.34i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.40118854910680213531869588928, −9.399681360999600534485312823501, −8.106945591929269602778703785671, −7.44646922200854895153280039676, −6.64549342136655720227987076547, −5.93959027485812574494640143846, −5.70384249274851340555170276341, −3.97480375547058893445506600229, −2.71904172186456383021501131664, −0.74111387569155173592928326953, 0.39526546445468173536740437684, 2.19234274889626766114642083316, 4.06278683425915206049909132479, 4.41164596421353440986658260128, 5.17664681644697809410963911620, 6.47950624220703606701918181524, 7.08280706329740275239320729055, 8.767592322965196896714420446890, 9.500784608134949486650078091824, 9.721254394260312860717262102906

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