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Properties

Label 2-950-95.53-c1-0-17
Degree $2$
Conductor $950$
Sign $0.822 - 0.569i$
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  + (0.422 + 0.906i)2-s + (0.284 + 0.199i)3-s + (−0.642 + 0.766i)4-s + (−0.0603 + 0.342i)6-s + (−0.814 − 3.03i)7-s + (−0.965 − 0.258i)8-s + (−0.984 − 2.70i)9-s + (1.15 + 2.00i)11-s + (−0.335 + 0.0898i)12-s + (3.68 + 5.26i)13-s + (2.40 − 2.02i)14-s + (−0.173 − 0.984i)16-s + (0.217 − 0.101i)17-s + (2.03 − 2.03i)18-s + (4.33 + 0.449i)19-s + ⋯
L(s)  = 1  + (0.298 + 0.640i)2-s + (0.164 + 0.115i)3-s + (−0.321 + 0.383i)4-s + (−0.0246 + 0.139i)6-s + (−0.307 − 1.14i)7-s + (−0.341 − 0.0915i)8-s + (−0.328 − 0.901i)9-s + (0.349 + 0.604i)11-s + (−0.0968 + 0.0259i)12-s + (1.02 + 1.46i)13-s + (0.643 − 0.540i)14-s + (−0.0434 − 0.246i)16-s + (0.0527 − 0.0246i)17-s + (0.479 − 0.479i)18-s + (0.994 + 0.103i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 - 0.569i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.822 - 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.822 - 0.569i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.822 - 0.569i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.84060 + 0.574871i\)
\(L(\frac12)\) \(\approx\) \(1.84060 + 0.574871i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.422 - 0.906i)T \)
5 \( 1 \)
19 \( 1 + (-4.33 - 0.449i)T \)
good3 \( 1 + (-0.284 - 0.199i)T + (1.02 + 2.81i)T^{2} \)
7 \( 1 + (0.814 + 3.03i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-1.15 - 2.00i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.68 - 5.26i)T + (-4.44 + 12.2i)T^{2} \)
17 \( 1 + (-0.217 + 0.101i)T + (10.9 - 13.0i)T^{2} \)
23 \( 1 + (0.265 + 3.03i)T + (-22.6 + 3.99i)T^{2} \)
29 \( 1 + (-9.33 + 3.39i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-7.73 - 4.46i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.25 + 2.25i)T + 37iT^{2} \)
41 \( 1 + (-11.4 + 2.01i)T + (38.5 - 14.0i)T^{2} \)
43 \( 1 + (0.196 + 0.0171i)T + (42.3 + 7.46i)T^{2} \)
47 \( 1 + (1.58 - 3.38i)T + (-30.2 - 36.0i)T^{2} \)
53 \( 1 + (-0.625 + 0.0547i)T + (52.1 - 9.20i)T^{2} \)
59 \( 1 + (9.92 + 3.61i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (7.07 + 5.93i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (10.5 + 4.90i)T + (43.0 + 51.3i)T^{2} \)
71 \( 1 + (-1.44 - 1.72i)T + (-12.3 + 69.9i)T^{2} \)
73 \( 1 + (8.79 - 12.5i)T + (-24.9 - 68.5i)T^{2} \)
79 \( 1 + (1.43 + 8.13i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-2.76 + 0.739i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (2.79 - 15.8i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-0.158 - 0.340i)T + (-62.3 + 74.3i)T^{2} \)
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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

−9.911395315389453717732986105612, −9.237621524940727298900287850883, −8.449647472510544514203335085213, −7.40562295071132510423916451352, −6.58522047720400920633169853228, −6.18473494540700040153452564870, −4.56657189377186289932693737529, −4.07589972229555139601224154484, −3.04106179632174367631526808265, −1.07647715999942824501113832251, 1.17220700232847279766037649515, 2.78284026542469631687707350838, 3.15002918156365817277840035609, 4.67429556995627797954440594654, 5.71207084932649367640622974147, 6.07733602246223721260222392848, 7.67918743878893177139193667879, 8.454388461241423818711221411501, 9.065044577318220802153537314554, 10.13213915454717881621809218071

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