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Properties

Label 2-950-19.17-c1-0-22
Degree $2$
Conductor $950$
Sign $0.533 + 0.845i$
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  + (0.173 − 0.984i)2-s + (1.28 + 1.07i)3-s + (−0.939 − 0.342i)4-s + (1.28 − 1.07i)6-s + (0.237 + 0.411i)7-s + (−0.5 + 0.866i)8-s + (−0.0320 − 0.181i)9-s + (2.79 − 4.84i)11-s + (−0.838 − 1.45i)12-s + (−1.88 + 1.58i)13-s + (0.446 − 0.162i)14-s + (0.766 + 0.642i)16-s + (0.498 − 2.82i)17-s − 0.184·18-s + (3.17 − 2.98i)19-s + ⋯
L(s)  = 1  + (0.122 − 0.696i)2-s + (0.742 + 0.622i)3-s + (−0.469 − 0.171i)4-s + (0.524 − 0.440i)6-s + (0.0898 + 0.155i)7-s + (−0.176 + 0.306i)8-s + (−0.0106 − 0.0606i)9-s + (0.843 − 1.46i)11-s + (−0.242 − 0.419i)12-s + (−0.523 + 0.438i)13-s + (0.119 − 0.0434i)14-s + (0.191 + 0.160i)16-s + (0.121 − 0.686i)17-s − 0.0435·18-s + (0.729 − 0.683i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.533 + 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.533 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.533 + 0.845i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.533 + 0.845i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86374 - 1.02784i\)
\(L(\frac12)\) \(\approx\) \(1.86374 - 1.02784i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 + 0.984i)T \)
5 \( 1 \)
19 \( 1 + (-3.17 + 2.98i)T \)
good3 \( 1 + (-1.28 - 1.07i)T + (0.520 + 2.95i)T^{2} \)
7 \( 1 + (-0.237 - 0.411i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.79 + 4.84i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.88 - 1.58i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-0.498 + 2.82i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (-4.12 - 1.50i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-1.51 - 8.59i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (2.88 + 4.99i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.02T + 37T^{2} \)
41 \( 1 + (-5.18 - 4.34i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-0.433 + 0.157i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (-0.456 - 2.59i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (7.43 + 2.70i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (1.42 - 8.06i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (2.98 + 1.08i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (0.0262 + 0.148i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (4.48 - 1.63i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (7.04 + 5.91i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (-9.48 - 7.96i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (6.59 + 11.4i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.23 - 1.03i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (0.873 - 4.95i)T + (-91.1 - 33.1i)T^{2} \)
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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

−9.646170552450668513041344314065, −9.185866025502806505619229682685, −8.714087659834392463322317589011, −7.53822087018267815853406350450, −6.40017363518695581892195956880, −5.30383667483258978675419443439, −4.32586573830349689346487579501, −3.33362500873495280323779117236, −2.75819938596764378565323232547, −1.01868266962988604268056867175, 1.48888947219484950749194933336, 2.71442258197814918729088589971, 4.01712329750533566481744696342, 4.90885083392617781023320062615, 6.03016879250686875721312688272, 7.06031261586351005623584308921, 7.56980114273584130589246411395, 8.236001418760688128880892490030, 9.239779397061291997299509967176, 9.870199327059711225640105604004

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