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Properties

Label 2-950-25.11-c1-0-22
Degree $2$
Conductor $950$
Sign $0.983 + 0.182i$
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 + (−0.0472 + 0.145i)3-s + (0.309 − 0.951i)4-s + (1.99 − 1.01i)5-s + (−0.0472 − 0.145i)6-s − 0.777·7-s + (0.309 + 0.951i)8-s + (2.40 + 1.74i)9-s + (−1.01 + 1.99i)10-s + (4.73 − 3.43i)11-s + (0.123 + 0.0898i)12-s + (−0.561 − 0.408i)13-s + (0.629 − 0.457i)14-s + (0.0542 + 0.337i)15-s + (−0.809 − 0.587i)16-s + (−1.47 − 4.54i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (−0.0272 + 0.0839i)3-s + (0.154 − 0.475i)4-s + (0.889 − 0.455i)5-s + (−0.0192 − 0.0593i)6-s − 0.293·7-s + (0.109 + 0.336i)8-s + (0.802 + 0.583i)9-s + (−0.319 + 0.630i)10-s + (1.42 − 1.03i)11-s + (0.0357 + 0.0259i)12-s + (−0.155 − 0.113i)13-s + (0.168 − 0.122i)14-s + (0.0140 + 0.0871i)15-s + (−0.202 − 0.146i)16-s + (−0.358 − 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.182i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.983 + 0.182i$
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: \((2,\ 950,\ (\ :1/2),\ 0.983 + 0.182i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49298 - 0.137751i\)
\(L(\frac12)\) \(\approx\) \(1.49298 - 0.137751i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.99 + 1.01i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
good3 \( 1 + (0.0472 - 0.145i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + 0.777T + 7T^{2} \)
11 \( 1 + (-4.73 + 3.43i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (0.561 + 0.408i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.47 + 4.54i)T + (-13.7 + 9.99i)T^{2} \)
23 \( 1 + (4.59 - 3.34i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.04 + 3.22i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.92 - 5.92i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (4.26 + 3.09i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (3.26 + 2.36i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 10.9T + 43T^{2} \)
47 \( 1 + (-3.20 + 9.85i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (0.365 - 1.12i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-10.4 - 7.60i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-1.41 + 1.02i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (1.59 + 4.90i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (3.33 - 10.2i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-3.20 + 2.32i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-3.19 + 9.82i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-2.04 - 6.29i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-10.2 + 7.47i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (2.70 - 8.31i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.879757174119563341438267901835, −9.152844854536023765489502269958, −8.583209706999125231935639704058, −7.43522863263383622940058360123, −6.62093576586672721940598973078, −5.83141166771487775042399569500, −4.95445154887762467136014925592, −3.76985414318179608784376499974, −2.17927570643455968659153757388, −0.994539049079922737832473006072, 1.38012481357841025254169548001, 2.25772422259709544061509600423, 3.68819047509032380423302406539, 4.50372986269841753515869266398, 6.21850884811786296549143710335, 6.58812557079858127581271811877, 7.43953074538780754110481941680, 8.676507270760510238879187265229, 9.532874823679998930967719666973, 9.860194986398365583593637053424

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