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Properties

Label 2-850-85.23-c1-0-5
Degree $2$
Conductor $850$
Sign $0.637 - 0.770i$
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 + (−1.41 − 0.942i)3-s + (0.707 − 0.707i)4-s + (1.66 + 0.330i)6-s + (5.02 + 0.999i)7-s + (−0.382 + 0.923i)8-s + (−0.0471 − 0.113i)9-s + (−0.999 + 5.02i)11-s + (−1.66 + 0.330i)12-s + 3.13·13-s + (−5.02 + 0.999i)14-s − i·16-s + (−3.59 + 2.02i)17-s + (0.0871 + 0.0871i)18-s + (−0.297 + 0.718i)19-s + ⋯
L(s)  = 1  + (−0.653 + 0.270i)2-s + (−0.814 − 0.544i)3-s + (0.353 − 0.353i)4-s + (0.679 + 0.135i)6-s + (1.89 + 0.377i)7-s + (−0.135 + 0.326i)8-s + (−0.0157 − 0.0379i)9-s + (−0.301 + 1.51i)11-s + (−0.480 + 0.0955i)12-s + 0.870·13-s + (−1.34 + 0.267i)14-s − 0.250i·16-s + (−0.871 + 0.490i)17-s + (0.0205 + 0.0205i)18-s + (−0.0682 + 0.164i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(850\)    =    \(2 \cdot 5^{2} \cdot 17\)
Sign: $0.637 - 0.770i$
Analytic conductor: \(6.78728\)
Root analytic conductor: \(2.60524\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{850} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 850,\ (\ :1/2),\ 0.637 - 0.770i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.863166 + 0.406084i\)
\(L(\frac12)\) \(\approx\) \(0.863166 + 0.406084i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.59 - 2.02i)T \)
good3 \( 1 + (1.41 + 0.942i)T + (1.14 + 2.77i)T^{2} \)
7 \( 1 + (-5.02 - 0.999i)T + (6.46 + 2.67i)T^{2} \)
11 \( 1 + (0.999 - 5.02i)T + (-10.1 - 4.20i)T^{2} \)
13 \( 1 - 3.13T + 13T^{2} \)
19 \( 1 + (0.297 - 0.718i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (0.327 + 0.489i)T + (-8.80 + 21.2i)T^{2} \)
29 \( 1 + (1.78 - 2.66i)T + (-11.0 - 26.7i)T^{2} \)
31 \( 1 + (0.00940 + 0.0472i)T + (-28.6 + 11.8i)T^{2} \)
37 \( 1 + (-1.22 + 1.83i)T + (-14.1 - 34.1i)T^{2} \)
41 \( 1 + (-0.774 - 1.15i)T + (-15.6 + 37.8i)T^{2} \)
43 \( 1 + (2.55 + 1.05i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 1.90iT - 47T^{2} \)
53 \( 1 + (-4.43 - 10.6i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-2.75 + 1.14i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (8.20 - 5.48i)T + (23.3 - 56.3i)T^{2} \)
67 \( 1 + (-7.07 - 7.07i)T + 67iT^{2} \)
71 \( 1 + (-7.55 + 1.50i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (7.03 - 1.39i)T + (67.4 - 27.9i)T^{2} \)
79 \( 1 + (-9.46 - 1.88i)T + (72.9 + 30.2i)T^{2} \)
83 \( 1 + (-15.6 + 6.48i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-5.07 + 5.07i)T - 89iT^{2} \)
97 \( 1 + (1.77 - 0.353i)T + (89.6 - 37.1i)T^{2} \)
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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

−10.60121926053717449205217242800, −9.275996157774977522275234134958, −8.528721685681358941438101621186, −7.71989102152843851387737312590, −6.99528342544667364460477575504, −6.02096759732494127148846282996, −5.18988598613300056556348426949, −4.29410555821626699760728847639, −2.13500671523746053300051050288, −1.33151420697606896818492962723, 0.71803945126421930112274718240, 2.14258752055984169883396487528, 3.74072341682250584046313996626, 4.80272419705213370071599262041, 5.53314417873792835756086463903, 6.57633861813023007818487222384, 7.954722145459064834193418693674, 8.258629397643026973100692061127, 9.184332951834521324475775178483, 10.46310833408983131513373863805

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