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Properties

Label 2-950-19.6-c1-0-3
Degree $2$
Conductor $950$
Sign $0.757 - 0.652i$
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  + (0.766 − 0.642i)2-s + (−2.78 − 1.01i)3-s + (0.173 − 0.984i)4-s + (−2.78 + 1.01i)6-s + (−0.454 + 0.787i)7-s + (−0.500 − 0.866i)8-s + (4.44 + 3.72i)9-s + (−1.82 − 3.16i)11-s + (−1.48 + 2.56i)12-s + (−2.66 + 0.969i)13-s + (0.157 + 0.895i)14-s + (−0.939 − 0.342i)16-s + (−4.13 + 3.46i)17-s + 5.80·18-s + (3.41 + 2.70i)19-s + ⋯
L(s)  = 1  + (0.541 − 0.454i)2-s + (−1.60 − 0.585i)3-s + (0.0868 − 0.492i)4-s + (−1.13 + 0.414i)6-s + (−0.171 + 0.297i)7-s + (−0.176 − 0.306i)8-s + (1.48 + 1.24i)9-s + (−0.550 − 0.953i)11-s + (−0.428 + 0.741i)12-s + (−0.738 + 0.268i)13-s + (0.0422 + 0.239i)14-s + (−0.234 − 0.0855i)16-s + (−1.00 + 0.840i)17-s + 1.36·18-s + (0.783 + 0.621i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 - 0.652i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.757 - 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.757 - 0.652i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.757 - 0.652i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.559715 + 0.207684i\)
\(L(\frac12)\) \(\approx\) \(0.559715 + 0.207684i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 + 0.642i)T \)
5 \( 1 \)
19 \( 1 + (-3.41 - 2.70i)T \)
good3 \( 1 + (2.78 + 1.01i)T + (2.29 + 1.92i)T^{2} \)
7 \( 1 + (0.454 - 0.787i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.82 + 3.16i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.66 - 0.969i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (4.13 - 3.46i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (0.226 - 1.28i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-3.16 - 2.65i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (2.98 - 5.16i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 9.82T + 37T^{2} \)
41 \( 1 + (0.0389 + 0.0141i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.54 + 8.74i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (1.74 + 1.46i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (1.76 - 10.0i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (-4.30 + 3.61i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (0.646 - 3.66i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-10.4 - 8.80i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-1.01 - 5.75i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-13.2 - 4.81i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (14.0 + 5.12i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-2.39 + 4.15i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (8.41 - 3.06i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (12.7 - 10.7i)T + (16.8 - 95.5i)T^{2} \)
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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

−10.58734502535581360179119754977, −9.617225382818066228541832502706, −8.403350493785667386909416214824, −7.25564139962957070077690599547, −6.47170145513863721944808870675, −5.67379809288024480123446869586, −5.16347117290611571419418327429, −4.02753963626848212419526448022, −2.57305540378129133708492531785, −1.20750333982862096971410880881, 0.31420028891764585514516296319, 2.60006395632561232747499965084, 4.18890675756335748879008792186, 4.82220104249346256383666932013, 5.39304403015408308216386020790, 6.47906547078775358635984044369, 7.04246739431533031153172618111, 7.945735941519304040623767709595, 9.641792361775080342071637523503, 9.795256155939897231554375379397

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