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Properties

Label 2-950-25.16-c1-0-9
Degree $2$
Conductor $950$
Sign $0.525 + 0.850i$
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  + (−0.809 − 0.587i)2-s + (−0.474 − 1.45i)3-s + (0.309 + 0.951i)4-s + (−1.96 + 1.06i)5-s + (−0.474 + 1.45i)6-s − 4.21·7-s + (0.309 − 0.951i)8-s + (0.523 − 0.380i)9-s + (2.21 + 0.293i)10-s + (3.04 + 2.21i)11-s + (1.24 − 0.901i)12-s + (−0.801 + 0.582i)13-s + (3.40 + 2.47i)14-s + (2.48 + 2.36i)15-s + (−0.809 + 0.587i)16-s + (−0.611 + 1.88i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (−0.273 − 0.842i)3-s + (0.154 + 0.475i)4-s + (−0.879 + 0.476i)5-s + (−0.193 + 0.595i)6-s − 1.59·7-s + (0.109 − 0.336i)8-s + (0.174 − 0.126i)9-s + (0.700 + 0.0928i)10-s + (0.917 + 0.666i)11-s + (0.358 − 0.260i)12-s + (−0.222 + 0.161i)13-s + (0.911 + 0.662i)14-s + (0.641 + 0.610i)15-s + (−0.202 + 0.146i)16-s + (−0.148 + 0.456i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.525 + 0.850i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.525 + 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.594579 - 0.331574i\)
\(L(\frac12)\) \(\approx\) \(0.594579 - 0.331574i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (1.96 - 1.06i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
good3 \( 1 + (0.474 + 1.45i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + 4.21T + 7T^{2} \)
11 \( 1 + (-3.04 - 2.21i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.801 - 0.582i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.611 - 1.88i)T + (-13.7 - 9.99i)T^{2} \)
23 \( 1 + (-3.18 - 2.31i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (0.752 + 2.31i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.0307 + 0.0947i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-2.08 + 1.51i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-6.25 + 4.54i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 4.77T + 43T^{2} \)
47 \( 1 + (1.72 + 5.29i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (2.09 + 6.44i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-2.24 + 1.63i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-9.21 - 6.69i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-0.902 + 2.77i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (3.31 + 10.1i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-12.1 - 8.81i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-0.300 - 0.925i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (1.49 - 4.59i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-3.97 - 2.88i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-3.95 - 12.1i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.758642685238413966918393743412, −9.319920017944760677147101450058, −8.152999650870252350702574041385, −7.06383897448992876814729468592, −6.93212842722959163745941804992, −6.01669818703656254311468354381, −4.13415038488070492988190612373, −3.48461386798367469862781486792, −2.20885027173300958580873803945, −0.66750560658890733589940664907, 0.74533060193713010119153865185, 3.02351345096063193813156842429, 3.98307249074202260820977985541, 4.87206027936289661373474314652, 5.98865199138396244154157982736, 6.82139638815066377927301547058, 7.62518023304879326121095183304, 8.803266095715916690108820727069, 9.301626460214771562025299829955, 9.946198344358659080556793707959

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