↑

Properties

Label 2-950-95.44-c1-0-13
Degree $2$
Conductor $950$
Sign $0.550 + 0.834i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.642 − 0.766i)2-s + (0.642 − 1.76i)3-s + (−0.173 + 0.984i)4-s + (−1.76 + 0.642i)6-s + (0.300 + 0.173i)7-s + (0.866 − 0.500i)8-s + (−0.407 − 0.342i)9-s + (2.97 + 5.14i)11-s + (1.62 + 0.939i)12-s + (−1.12 − 3.09i)13-s + (−0.0603 − 0.342i)14-s + (−0.939 − 0.342i)16-s + (3.55 + 4.23i)17-s + 0.532i·18-s + (4.11 + 1.43i)19-s + ⋯
L(s)  = 1  + (−0.454 − 0.541i)2-s + (0.371 − 1.01i)3-s + (−0.0868 + 0.492i)4-s + (−0.720 + 0.262i)6-s + (0.113 + 0.0656i)7-s + (0.306 − 0.176i)8-s + (−0.135 − 0.114i)9-s + (0.896 + 1.55i)11-s + (0.469 + 0.271i)12-s + (−0.312 − 0.857i)13-s + (−0.0161 − 0.0914i)14-s + (−0.234 − 0.0855i)16-s + (0.862 + 1.02i)17-s + 0.125i·18-s + (0.944 + 0.328i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.550 + 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.550 + 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.550 + 0.834i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (899, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.550 + 0.834i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39290 - 0.750041i\)
\(L(\frac12)\) \(\approx\) \(1.39290 - 0.750041i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.642 + 0.766i)T \)
5 \( 1 \)
19 \( 1 + (-4.11 - 1.43i)T \)
good3 \( 1 + (-0.642 + 1.76i)T + (-2.29 - 1.92i)T^{2} \)
7 \( 1 + (-0.300 - 0.173i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.97 - 5.14i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.12 + 3.09i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-3.55 - 4.23i)T + (-2.95 + 16.7i)T^{2} \)
23 \( 1 + (-2.79 - 0.492i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (5.46 + 4.58i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (1.52 - 2.63i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.82iT - 37T^{2} \)
41 \( 1 + (-6.41 - 2.33i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-0.524 + 0.0923i)T + (40.4 - 14.7i)T^{2} \)
47 \( 1 + (-2.99 + 3.56i)T + (-8.16 - 46.2i)T^{2} \)
53 \( 1 + (3.05 + 0.539i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (-2.10 + 1.76i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (2.14 - 12.1i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (3.24 - 3.86i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (2.26 + 12.8i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-0.752 + 2.06i)T + (-55.9 - 46.9i)T^{2} \)
79 \( 1 + (0.939 + 0.342i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (8.10 + 4.67i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-10.6 + 3.87i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-3.67 - 4.38i)T + (-16.8 + 95.5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.889266444766622843027637686575, −9.161833343889293531792697277148, −8.044759924825862521394280378499, −7.55969400533777400363347871072, −6.88090807917276600359519879707, −5.65334102918029159612873481466, −4.39945224516609678293459530629, −3.24937245030717848006101160215, −2.02422946351119220980957834132, −1.24586003093225916237484213025, 1.06599359249480202992786656821, 3.02443840169703720434673650714, 3.90212178734722593861175384552, 4.95915902792994826155151532940, 5.81901718008371339489640552714, 6.90241185496709974740889234837, 7.67515144283037294218010202993, 8.860836716285626410536071063866, 9.261786455919489922004487274269, 9.766771882646306495465935054886

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