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Properties

Label 2-950-25.21-c1-0-1
Degree $2$
Conductor $950$
Sign $0.788 - 0.615i$
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 + (−1.31 − 0.953i)3-s + (−0.809 − 0.587i)4-s + (−2.03 − 0.916i)5-s + (−1.31 + 0.953i)6-s − 4.02·7-s + (−0.809 + 0.587i)8-s + (−0.113 − 0.348i)9-s + (−1.50 + 1.65i)10-s + (0.355 − 1.09i)11-s + (0.501 + 1.54i)12-s + (−0.757 − 2.33i)13-s + (−1.24 + 3.82i)14-s + (1.80 + 3.14i)15-s + (0.309 + 0.951i)16-s + (−0.552 + 0.401i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (−0.758 − 0.550i)3-s + (−0.404 − 0.293i)4-s + (−0.912 − 0.410i)5-s + (−0.536 + 0.389i)6-s − 1.52·7-s + (−0.286 + 0.207i)8-s + (−0.0377 − 0.116i)9-s + (−0.475 + 0.523i)10-s + (0.107 − 0.330i)11-s + (0.144 + 0.445i)12-s + (−0.209 − 0.646i)13-s + (−0.332 + 1.02i)14-s + (0.465 + 0.813i)15-s + (0.0772 + 0.237i)16-s + (−0.133 + 0.0973i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.788 - 0.615i$
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: \((2,\ 950,\ (\ :1/2),\ 0.788 - 0.615i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0540566 + 0.0186070i\)
\(L(\frac12)\) \(\approx\) \(0.0540566 + 0.0186070i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.03 + 0.916i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
good3 \( 1 + (1.31 + 0.953i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + 4.02T + 7T^{2} \)
11 \( 1 + (-0.355 + 1.09i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (0.757 + 2.33i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.552 - 0.401i)T + (5.25 - 16.1i)T^{2} \)
23 \( 1 + (-0.354 + 1.09i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-1.48 - 1.08i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-3.11 + 2.26i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.546 - 1.68i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.29 - 3.97i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 1.89T + 43T^{2} \)
47 \( 1 + (3.82 + 2.77i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-3.51 - 2.55i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.0654 - 0.201i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (0.0373 - 0.114i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (10.4 - 7.60i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (11.3 + 8.25i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-4.32 + 13.3i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-5.95 - 4.32i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (0.615 - 0.447i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (0.940 - 2.89i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (0.410 + 0.298i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.26681001611467513460426230211, −9.401116833687175999034334101307, −8.602084702002985852708952496041, −7.49525080097379076261141015036, −6.55309388489163479340190891364, −5.88563907058216745045914891212, −4.80134975496116971633031475947, −3.63268547734916658432825752536, −2.92365191522412958719505316574, −0.940703899547448370065049272310, 0.03603050849766890478139912653, 2.84399982697889272244344192587, 3.89061996876522264983867048397, 4.63062158594764562348084853304, 5.70930159333786223469208901485, 6.58460545420683137155558297397, 7.12840116527898195788456521875, 8.137643323932434857426961698610, 9.209091002533356410467967931805, 9.961886364617899974585584942215

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