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Properties

Label 2-850-85.82-c1-0-25
Degree $2$
Conductor $850$
Sign $-0.509 + 0.860i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.923 + 0.382i)2-s + (−0.530 − 0.793i)3-s + (0.707 + 0.707i)4-s + (−0.186 − 0.935i)6-s + (−0.736 − 3.70i)7-s + (0.382 + 0.923i)8-s + (0.799 − 1.93i)9-s + (−2.74 + 0.546i)11-s + (0.186 − 0.935i)12-s − 5.74·13-s + (0.736 − 3.70i)14-s + i·16-s + (−3.33 − 2.41i)17-s + (1.47 − 1.47i)18-s + (0.119 + 0.288i)19-s + ⋯
L(s)  = 1  + (0.653 + 0.270i)2-s + (−0.306 − 0.458i)3-s + (0.353 + 0.353i)4-s + (−0.0760 − 0.382i)6-s + (−0.278 − 1.39i)7-s + (0.135 + 0.326i)8-s + (0.266 − 0.643i)9-s + (−0.828 + 0.164i)11-s + (0.0537 − 0.270i)12-s − 1.59·13-s + (0.196 − 0.989i)14-s + 0.250i·16-s + (−0.809 − 0.586i)17-s + (0.348 − 0.348i)18-s + (0.0273 + 0.0660i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.509 + 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.509 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(850\)    =    \(2 \cdot 5^{2} \cdot 17\)
Sign: $-0.509 + 0.860i$
Analytic conductor: \(6.78728\)
Root analytic conductor: \(2.60524\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{850} (507, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 850,\ (\ :1/2),\ -0.509 + 0.860i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.610974 - 1.07243i\)
\(L(\frac12)\) \(\approx\) \(0.610974 - 1.07243i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.923 - 0.382i)T \)
5 \( 1 \)
17 \( 1 + (3.33 + 2.41i)T \)
good3 \( 1 + (0.530 + 0.793i)T + (-1.14 + 2.77i)T^{2} \)
7 \( 1 + (0.736 + 3.70i)T + (-6.46 + 2.67i)T^{2} \)
11 \( 1 + (2.74 - 0.546i)T + (10.1 - 4.20i)T^{2} \)
13 \( 1 + 5.74T + 13T^{2} \)
19 \( 1 + (-0.119 - 0.288i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (-3.97 - 2.65i)T + (8.80 + 21.2i)T^{2} \)
29 \( 1 + (-0.463 + 0.309i)T + (11.0 - 26.7i)T^{2} \)
31 \( 1 + (5.00 + 0.994i)T + (28.6 + 11.8i)T^{2} \)
37 \( 1 + (-8.56 + 5.72i)T + (14.1 - 34.1i)T^{2} \)
41 \( 1 + (7.70 + 5.15i)T + (15.6 + 37.8i)T^{2} \)
43 \( 1 + (-8.32 + 3.44i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 - 5.61iT - 47T^{2} \)
53 \( 1 + (-2.52 + 6.08i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-3.14 - 1.30i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (-5.67 + 8.48i)T + (-23.3 - 56.3i)T^{2} \)
67 \( 1 + (1.94 - 1.94i)T - 67iT^{2} \)
71 \( 1 + (-0.782 + 3.93i)T + (-65.5 - 27.1i)T^{2} \)
73 \( 1 + (-0.167 + 0.841i)T + (-67.4 - 27.9i)T^{2} \)
79 \( 1 + (0.872 + 4.38i)T + (-72.9 + 30.2i)T^{2} \)
83 \( 1 + (-1.36 - 0.565i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (-7.49 - 7.49i)T + 89iT^{2} \)
97 \( 1 + (-3.04 + 15.3i)T + (-89.6 - 37.1i)T^{2} \)
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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

−9.965692575567935888145506754377, −9.187391581330768265404139667874, −7.55504975086112087683345611955, −7.33913118774258279613577546589, −6.61754552371347711656516196507, −5.42299641530745659074735238721, −4.55903818714632528851569676965, −3.60248807865581712764979721009, −2.31211877845741735807875319646, −0.46344320223444250675836865542, 2.22612412342876828070418106358, 2.83590558810903969802851839778, 4.43347630457304102977826770628, 5.10413925523247121901106264235, 5.76555029380789679621879622492, 6.86276155048658872241105772435, 7.906480734458418218782270047463, 8.941471852678795167157901073615, 9.823749792457605927145128987725, 10.52118788088564593958563169982

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