↑

Properties

Label 2-850-85.12-c1-0-3
Degree $2$
Conductor $850$
Sign $-0.918 + 0.396i$
Analytic cond. $6.78728$
Root an. cond. $2.60524$
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  + (−0.382 + 0.923i)2-s + (−0.372 + 1.87i)3-s + (−0.707 − 0.707i)4-s + (−1.58 − 1.06i)6-s + (0.930 + 0.622i)7-s + (0.923 − 0.382i)8-s + (−0.596 − 0.247i)9-s + (−1.86 + 2.79i)11-s + (1.58 − 1.06i)12-s + 1.47·13-s + (−0.930 + 0.622i)14-s + i·16-s + (−3.90 + 1.32i)17-s + (0.456 − 0.456i)18-s + (−4.07 + 1.68i)19-s + ⋯
L(s)  = 1  + (−0.270 + 0.653i)2-s + (−0.215 + 1.08i)3-s + (−0.353 − 0.353i)4-s + (−0.648 − 0.433i)6-s + (0.351 + 0.235i)7-s + (0.326 − 0.135i)8-s + (−0.198 − 0.0823i)9-s + (−0.562 + 0.842i)11-s + (0.458 − 0.306i)12-s + 0.409·13-s + (−0.248 + 0.166i)14-s + 0.250i·16-s + (−0.946 + 0.322i)17-s + (0.107 − 0.107i)18-s + (−0.934 + 0.386i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.918 + 0.396i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.918 + 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(850\)    =    \(2 \cdot 5^{2} \cdot 17\)
Sign: $-0.918 + 0.396i$
Analytic conductor: \(6.78728\)
Root analytic conductor: \(2.60524\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{850} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 850,\ (\ :1/2),\ -0.918 + 0.396i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.171611 - 0.830378i\)
\(L(\frac12)\) \(\approx\) \(0.171611 - 0.830378i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.382 - 0.923i)T \)
5 \( 1 \)
17 \( 1 + (3.90 - 1.32i)T \)
good3 \( 1 + (0.372 - 1.87i)T + (-2.77 - 1.14i)T^{2} \)
7 \( 1 + (-0.930 - 0.622i)T + (2.67 + 6.46i)T^{2} \)
11 \( 1 + (1.86 - 2.79i)T + (-4.20 - 10.1i)T^{2} \)
13 \( 1 - 1.47T + 13T^{2} \)
19 \( 1 + (4.07 - 1.68i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-2.92 + 0.581i)T + (21.2 - 8.80i)T^{2} \)
29 \( 1 + (-3.34 - 0.664i)T + (26.7 + 11.0i)T^{2} \)
31 \( 1 + (-2.94 - 4.41i)T + (-11.8 + 28.6i)T^{2} \)
37 \( 1 + (8.28 + 1.64i)T + (34.1 + 14.1i)T^{2} \)
41 \( 1 + (10.6 - 2.11i)T + (37.8 - 15.6i)T^{2} \)
43 \( 1 + (2.55 + 6.16i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 - 6.49iT - 47T^{2} \)
53 \( 1 + (-5.24 - 2.17i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (-0.784 + 1.89i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-0.266 - 1.34i)T + (-56.3 + 23.3i)T^{2} \)
67 \( 1 + (-7.14 + 7.14i)T - 67iT^{2} \)
71 \( 1 + (-1.30 + 0.874i)T + (27.1 - 65.5i)T^{2} \)
73 \( 1 + (-10.5 + 7.04i)T + (27.9 - 67.4i)T^{2} \)
79 \( 1 + (6.00 + 4.01i)T + (30.2 + 72.9i)T^{2} \)
83 \( 1 + (4.89 - 11.8i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (5.20 + 5.20i)T + 89iT^{2} \)
97 \( 1 + (-8.14 + 5.44i)T + (37.1 - 89.6i)T^{2} \)
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

−10.43917689779655176463884475097, −9.942662735310211758226382441430, −8.823233243804582810106721422184, −8.395034685197999485264642326131, −7.16774016508325371782183417107, −6.38857050764620712037757894607, −5.11854059658977747353921019134, −4.73674732517538653453143547820, −3.64883972648396767820110836577, −1.93536642243602029178715292055, 0.46179083225507098873941451878, 1.72119195383946792669663530080, 2.81986309136565615235299307770, 4.13644507921078765424932124404, 5.25386610596044527633538069625, 6.48450445400904076137782544217, 7.08538040358457892148401942445, 8.275239222781451963664594688728, 8.573793926897503081026415027171, 9.848806232209406997357720888474

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