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Properties

Label 2-950-25.11-c1-0-28
Degree $2$
Conductor $950$
Sign $0.239 + 0.970i$
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 + (0.869 − 2.67i)3-s + (0.309 − 0.951i)4-s + (2.22 + 0.220i)5-s + (0.869 + 2.67i)6-s + 0.0565·7-s + (0.309 + 0.951i)8-s + (−3.98 − 2.89i)9-s + (−1.92 + 1.12i)10-s + (−0.352 + 0.255i)11-s + (−2.27 − 1.65i)12-s + (4.20 + 3.05i)13-s + (−0.0457 + 0.0332i)14-s + (2.52 − 5.76i)15-s + (−0.809 − 0.587i)16-s + (−1.62 − 5.01i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (0.502 − 1.54i)3-s + (0.154 − 0.475i)4-s + (0.995 + 0.0987i)5-s + (0.355 + 1.09i)6-s + 0.0213·7-s + (0.109 + 0.336i)8-s + (−1.32 − 0.964i)9-s + (−0.610 + 0.357i)10-s + (−0.106 + 0.0771i)11-s + (−0.657 − 0.477i)12-s + (1.16 + 0.847i)13-s + (−0.0122 + 0.00888i)14-s + (0.652 − 1.48i)15-s + (−0.202 − 0.146i)16-s + (−0.394 − 1.21i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.239 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.239 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.239 + 0.970i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (761, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.239 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35582 - 1.06186i\)
\(L(\frac12)\) \(\approx\) \(1.35582 - 1.06186i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.22 - 0.220i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
good3 \( 1 + (-0.869 + 2.67i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 - 0.0565T + 7T^{2} \)
11 \( 1 + (0.352 - 0.255i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-4.20 - 3.05i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.62 + 5.01i)T + (-13.7 + 9.99i)T^{2} \)
23 \( 1 + (-5.63 + 4.09i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.16 + 3.58i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.31 + 4.04i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (3.12 + 2.27i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-5.17 - 3.75i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 4.51T + 43T^{2} \)
47 \( 1 + (1.93 - 5.94i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (3.35 - 10.3i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (1.73 + 1.26i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (4.59 - 3.33i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (1.34 + 4.15i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-3.82 + 11.7i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (0.601 - 0.437i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-4.76 + 14.6i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-3.99 - 12.2i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (4.66 - 3.39i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (2.77 - 8.53i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.359758867952095175981093565726, −9.072397737403214700977697681479, −8.102898594895655020457294505950, −7.31830645822976569424938666959, −6.47174830775384980734516025746, −6.15973581389290852575540265866, −4.80698124944569323634649640969, −2.92486558493896732100662319018, −1.99677891959726569334082438355, −1.01749764720176498108409015611, 1.57215281255041416506928738549, 3.03149945187052506336860959305, 3.63411651766160386833074529361, 4.89671248083181694617038410082, 5.67342672483512755831885632799, 6.82272789606235346254600341449, 8.383306393304404603837060489489, 8.660518036542123885338021466986, 9.494901111833715058213194420898, 10.16460879894000120100871224269

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