↑

Properties

Label 2-950-475.117-c1-0-13
Degree $2$
Conductor $950$
Sign $0.939 + 0.343i$
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  + (−0.390 − 0.920i)2-s + (−1.35 + 0.263i)3-s + (−0.694 + 0.719i)4-s + (−2.23 + 0.0944i)5-s + (0.771 + 1.14i)6-s + (0.483 − 1.80i)7-s + (0.933 + 0.358i)8-s + (−1.01 + 0.411i)9-s + (0.959 + 2.01i)10-s + (−1.23 − 0.263i)11-s + (0.751 − 1.15i)12-s + (1.27 − 0.156i)13-s + (−1.85 + 0.260i)14-s + (2.99 − 0.715i)15-s + (−0.0348 − 0.999i)16-s + (−3.65 + 6.08i)17-s + ⋯
L(s)  = 1  + (−0.276 − 0.650i)2-s + (−0.781 + 0.151i)3-s + (−0.347 + 0.359i)4-s + (−0.999 + 0.0422i)5-s + (0.314 + 0.466i)6-s + (0.182 − 0.682i)7-s + (0.330 + 0.126i)8-s + (−0.339 + 0.137i)9-s + (0.303 + 0.638i)10-s + (−0.373 − 0.0794i)11-s + (0.216 − 0.333i)12-s + (0.352 − 0.0433i)13-s + (−0.494 + 0.0695i)14-s + (0.774 − 0.184i)15-s + (−0.00872 − 0.249i)16-s + (−0.887 + 1.47i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.343i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $0.939 + 0.343i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ 0.939 + 0.343i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.573939 - 0.101677i\)
\(L(\frac12)\) \(\approx\) \(0.573939 - 0.101677i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.390 + 0.920i)T \)
5 \( 1 + (2.23 - 0.0944i)T \)
19 \( 1 + (2.20 + 3.75i)T \)
good3 \( 1 + (1.35 - 0.263i)T + (2.78 - 1.12i)T^{2} \)
7 \( 1 + (-0.483 + 1.80i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (1.23 + 0.263i)T + (10.0 + 4.47i)T^{2} \)
13 \( 1 + (-1.27 + 0.156i)T + (12.6 - 3.14i)T^{2} \)
17 \( 1 + (3.65 - 6.08i)T + (-7.98 - 15.0i)T^{2} \)
23 \( 1 + (1.03 - 3.38i)T + (-19.0 - 12.8i)T^{2} \)
29 \( 1 + (-0.619 + 2.48i)T + (-25.6 - 13.6i)T^{2} \)
31 \( 1 + (0.926 + 0.0974i)T + (30.3 + 6.44i)T^{2} \)
37 \( 1 + (2.35 + 4.62i)T + (-21.7 + 29.9i)T^{2} \)
41 \( 1 + (-9.92 + 0.346i)T + (40.9 - 2.86i)T^{2} \)
43 \( 1 + (-3.99 - 5.71i)T + (-14.7 + 40.4i)T^{2} \)
47 \( 1 + (-6.26 + 3.76i)T + (22.0 - 41.4i)T^{2} \)
53 \( 1 + (-0.209 - 12.0i)T + (-52.9 + 1.84i)T^{2} \)
59 \( 1 + (-8.45 + 5.28i)T + (25.8 - 53.0i)T^{2} \)
61 \( 1 + (-13.1 + 6.98i)T + (34.1 - 50.5i)T^{2} \)
67 \( 1 + (-4.37 + 5.03i)T + (-9.32 - 66.3i)T^{2} \)
71 \( 1 + (1.85 - 0.902i)T + (43.7 - 55.9i)T^{2} \)
73 \( 1 + (10.3 + 1.26i)T + (70.8 + 17.6i)T^{2} \)
79 \( 1 + (-2.20 + 3.27i)T + (-29.5 - 73.2i)T^{2} \)
83 \( 1 + (-9.30 - 11.4i)T + (-17.2 + 81.1i)T^{2} \)
89 \( 1 + (0.184 - 5.29i)T + (-88.7 - 6.20i)T^{2} \)
97 \( 1 + (10.6 - 9.28i)T + (13.4 - 96.0i)T^{2} \)
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

−10.45838539707669684686355987901, −9.151961398266876745777847437856, −8.337827253582764367588343533475, −7.65651673950302820185747177259, −6.61539747926759815403098185642, −5.55818171242515640119457578530, −4.38277876102482680957347199165, −3.86805042547343504177279884813, −2.45077978384874839522568293867, −0.70223521818083395362834873366, 0.57138986442216975392931850984, 2.58910231754899754468771888806, 4.07650670720095454849295918655, 5.05977150456309014450319718739, 5.79194809254331965725968788072, 6.74775525346527294700939372662, 7.43356851051440929130553792050, 8.608928006936312911674654932456, 8.777159290411264129479092362849, 10.13821772119260057582770590688

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