Properties

Label 2-950-19.7-c1-0-32
Degree $2$
Conductor $950$
Sign $-0.444 + 0.895i$
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  + (0.5 + 0.866i)2-s + (0.236 + 0.410i)3-s + (−0.499 + 0.866i)4-s + (−0.236 + 0.410i)6-s − 2.19·7-s − 0.999·8-s + (1.38 − 2.40i)9-s − 4.96·11-s − 0.473·12-s + (−1.14 + 1.97i)13-s + (−1.09 − 1.89i)14-s + (−0.5 − 0.866i)16-s + (−3.38 − 5.86i)17-s + 2.77·18-s + (−3.12 − 3.03i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.136 + 0.236i)3-s + (−0.249 + 0.433i)4-s + (−0.0967 + 0.167i)6-s − 0.828·7-s − 0.353·8-s + (0.462 − 0.801i)9-s − 1.49·11-s − 0.136·12-s + (−0.316 + 0.548i)13-s + (−0.292 − 0.507i)14-s + (−0.125 − 0.216i)16-s + (−0.821 − 1.42i)17-s + 0.654·18-s + (−0.716 − 0.697i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.444 + 0.895i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.444 + 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0727467 - 0.117330i\)
\(L(\frac12)\) \(\approx\) \(0.0727467 - 0.117330i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 \)
19 \( 1 + (3.12 + 3.03i)T \)
good3 \( 1 + (-0.236 - 0.410i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + 2.19T + 7T^{2} \)
11 \( 1 + 4.96T + 11T^{2} \)
13 \( 1 + (1.14 - 1.97i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.38 + 5.86i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (3.84 - 6.65i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.983 - 1.70i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.58T + 31T^{2} \)
37 \( 1 + 7.38T + 37T^{2} \)
41 \( 1 + (5.70 + 9.87i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.07 - 8.79i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.09 - 3.62i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.09 + 3.62i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.72 + 2.97i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.36 + 4.09i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.404 - 0.700i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.59 - 9.69i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.42 - 4.20i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.63 + 13.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.7T + 83T^{2} \)
89 \( 1 + (-1.78 + 3.08i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.0861 - 0.149i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.596457806263269981480348607047, −9.054518918389041391226240214602, −7.967739152886103222997485911331, −7.01917341010228167223628437842, −6.52805951676610424026340175700, −5.33408388970902385900517781350, −4.55749606981832165152669494730, −3.46425515266230373473441642647, −2.48911409034411732175461367504, −0.05097454634475016535305884233, 1.99795340670003467585805868442, 2.76308146997164563599962338671, 4.02127353296723285186274160380, 4.92128154287438073776579033435, 5.96633517405253414286495009952, 6.77691948686718057546693162871, 8.098502770486578672606129236102, 8.394819273173054095798266905320, 9.901713531293675661266869135649, 10.45359871539057914159870033037

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