Properties

Label 2-950-19.7-c1-0-20
Degree $2$
Conductor $950$
Sign $0.425 + 0.905i$
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 + (−1.32 − 2.29i)3-s + (−0.499 + 0.866i)4-s + (1.32 − 2.29i)6-s + 1.64·7-s − 0.999·8-s + (−2 + 3.46i)9-s + 0.645·11-s + 2.64·12-s + (1 − 1.73i)13-s + (0.822 + 1.42i)14-s + (−0.5 − 0.866i)16-s − 3.99·18-s + (4.32 − 0.559i)19-s + (−2.17 − 3.77i)21-s + (0.322 + 0.559i)22-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.763 − 1.32i)3-s + (−0.249 + 0.433i)4-s + (0.540 − 0.935i)6-s + 0.622·7-s − 0.353·8-s + (−0.666 + 1.15i)9-s + 0.194·11-s + 0.763·12-s + (0.277 − 0.480i)13-s + (0.219 + 0.380i)14-s + (−0.125 − 0.216i)16-s − 0.942·18-s + (0.991 − 0.128i)19-s + (−0.475 − 0.822i)21-s + (0.0688 + 0.119i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.425 + 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.425 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.425 + 0.905i$
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: \((2,\ 950,\ (\ :1/2),\ 0.425 + 0.905i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19578 - 0.759226i\)
\(L(\frac12)\) \(\approx\) \(1.19578 - 0.759226i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.32 + 0.559i)T \)
good3 \( 1 + (1.32 + 2.29i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 - 1.64T + 7T^{2} \)
11 \( 1 - 0.645T + 11T^{2} \)
13 \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-1.82 + 3.15i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.82 + 3.15i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.354T + 31T^{2} \)
37 \( 1 + 5.64T + 37T^{2} \)
41 \( 1 + (5.14 + 8.91i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.354 - 0.613i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.82 + 8.35i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.29 + 7.43i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.96 + 6.87i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.46 + 12.9i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.32 - 4.02i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.64 - 11.5i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-6.14 - 10.6i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.93T + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.14 + 12.3i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.921951953810308148172618349963, −8.613248851765468320222557668744, −8.022187785923927752655715676507, −7.09608817484488854108990640363, −6.64171961190774487787455054159, −5.56612440978367954838804267885, −5.06681677368995586296390771494, −3.62759530467967188805980545951, −2.10851796416773059397272283348, −0.72850088022051587148970526131, 1.39064012244530571073852300380, 3.11986061568923608599365974434, 4.04210654744120881818712851726, 4.87049677411462885341169887922, 5.44593324572150411507713806428, 6.45675491761333469288233375478, 7.73092213815635777742781317321, 9.007168791645991815462935503004, 9.487785954762172765953436665016, 10.46500046302908722988876762507

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