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Properties

Label 2-950-25.6-c1-0-0
Degree $2$
Conductor $950$
Sign $-0.169 - 0.985i$
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  + (−0.309 − 0.951i)2-s + (0.573 − 0.416i)3-s + (−0.809 + 0.587i)4-s + (−0.120 − 2.23i)5-s + (−0.573 − 0.416i)6-s − 2.96·7-s + (0.809 + 0.587i)8-s + (−0.771 + 2.37i)9-s + (−2.08 + 0.804i)10-s + (−0.913 − 2.81i)11-s + (−0.219 + 0.674i)12-s + (−0.396 + 1.22i)13-s + (0.917 + 2.82i)14-s + (−0.999 − 1.23i)15-s + (0.309 − 0.951i)16-s + (−1.57 − 1.14i)17-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (0.331 − 0.240i)3-s + (−0.404 + 0.293i)4-s + (−0.0538 − 0.998i)5-s + (−0.234 − 0.170i)6-s − 1.12·7-s + (0.286 + 0.207i)8-s + (−0.257 + 0.791i)9-s + (−0.659 + 0.254i)10-s + (−0.275 − 0.847i)11-s + (−0.0632 + 0.194i)12-s + (−0.110 + 0.338i)13-s + (0.245 + 0.754i)14-s + (−0.258 − 0.317i)15-s + (0.0772 − 0.237i)16-s + (−0.381 − 0.277i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.169 - 0.985i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (381, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.169 - 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0195690 + 0.0232272i\)
\(L(\frac12)\) \(\approx\) \(0.0195690 + 0.0232272i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (0.120 + 2.23i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
good3 \( 1 + (-0.573 + 0.416i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 + 2.96T + 7T^{2} \)
11 \( 1 + (0.913 + 2.81i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (0.396 - 1.22i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.57 + 1.14i)T + (5.25 + 16.1i)T^{2} \)
23 \( 1 + (-1.95 - 6.00i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (7.49 - 5.44i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.67 - 1.21i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.64 + 8.15i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.13 - 6.57i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 5.90T + 43T^{2} \)
47 \( 1 + (2.65 - 1.92i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-9.24 + 6.71i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.96 - 9.12i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.85 + 5.71i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-5.38 - 3.90i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (9.15 - 6.65i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (3.37 + 10.3i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-0.476 + 0.346i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (11.8 + 8.64i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-0.364 - 1.12i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-2.40 + 1.74i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.19071699743980528938392978691, −9.259766127910502582633310939887, −8.873332064679783594524150334517, −7.962452906932480688439524460636, −7.10143362683945871514270220927, −5.77834072654497031720549785418, −4.98980291321308209439216057899, −3.74916437918260975141248590228, −2.85201789508963742206385852090, −1.59618570745482266984013645650, 0.01402819301093179428401851285, 2.45083175778225853237153400472, 3.43197228475743994200517937301, 4.37054867505223997108574024621, 5.81319112904469767561567358795, 6.53510318167078592394256716505, 7.08501598806208163015627910678, 8.103561152317764012521359921072, 9.004417936671503747180003806040, 9.951001808930620418685282455382

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