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Properties

Label 2-850-85.28-c1-0-6
Degree $2$
Conductor $850$
Sign $-0.719 - 0.694i$
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 + (−0.776 + 1.16i)3-s + (0.707 − 0.707i)4-s + (0.272 − 1.37i)6-s + (−0.749 + 3.77i)7-s + (−0.382 + 0.923i)8-s + (0.399 + 0.965i)9-s + (3.05 + 0.606i)11-s + (0.272 + 1.37i)12-s + 4.47·13-s + (−0.749 − 3.77i)14-s − i·16-s + (2.57 − 3.21i)17-s + (−0.738 − 0.738i)18-s + (−1.25 + 3.02i)19-s + ⋯
L(s)  = 1  + (−0.653 + 0.270i)2-s + (−0.448 + 0.671i)3-s + (0.353 − 0.353i)4-s + (0.111 − 0.559i)6-s + (−0.283 + 1.42i)7-s + (−0.135 + 0.326i)8-s + (0.133 + 0.321i)9-s + (0.919 + 0.182i)11-s + (0.0787 + 0.395i)12-s + 1.24·13-s + (−0.200 − 1.00i)14-s − 0.250i·16-s + (0.625 − 0.780i)17-s + (−0.174 − 0.174i)18-s + (−0.287 + 0.692i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.719 - 0.694i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.719 - 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(850\)    =    \(2 \cdot 5^{2} \cdot 17\)
Sign: $-0.719 - 0.694i$
Analytic conductor: \(6.78728\)
Root analytic conductor: \(2.60524\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{850} (793, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 850,\ (\ :1/2),\ -0.719 - 0.694i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.368960 + 0.913463i\)
\(L(\frac12)\) \(\approx\) \(0.368960 + 0.913463i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.57 + 3.21i)T \)
good3 \( 1 + (0.776 - 1.16i)T + (-1.14 - 2.77i)T^{2} \)
7 \( 1 + (0.749 - 3.77i)T + (-6.46 - 2.67i)T^{2} \)
11 \( 1 + (-3.05 - 0.606i)T + (10.1 + 4.20i)T^{2} \)
13 \( 1 - 4.47T + 13T^{2} \)
19 \( 1 + (1.25 - 3.02i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-2.05 + 1.36i)T + (8.80 - 21.2i)T^{2} \)
29 \( 1 + (-5.13 - 3.42i)T + (11.0 + 26.7i)T^{2} \)
31 \( 1 + (6.04 - 1.20i)T + (28.6 - 11.8i)T^{2} \)
37 \( 1 + (6.31 + 4.21i)T + (14.1 + 34.1i)T^{2} \)
41 \( 1 + (-1.67 + 1.12i)T + (15.6 - 37.8i)T^{2} \)
43 \( 1 + (-5.82 - 2.41i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 9.31iT - 47T^{2} \)
53 \( 1 + (-4.76 - 11.5i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (4.66 - 1.93i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (1.54 + 2.31i)T + (-23.3 + 56.3i)T^{2} \)
67 \( 1 + (6.06 + 6.06i)T + 67iT^{2} \)
71 \( 1 + (-0.137 - 0.691i)T + (-65.5 + 27.1i)T^{2} \)
73 \( 1 + (0.264 + 1.32i)T + (-67.4 + 27.9i)T^{2} \)
79 \( 1 + (-2.06 + 10.3i)T + (-72.9 - 30.2i)T^{2} \)
83 \( 1 + (12.3 - 5.12i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (4.33 - 4.33i)T - 89iT^{2} \)
97 \( 1 + (-1.53 - 7.72i)T + (-89.6 + 37.1i)T^{2} \)
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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

−10.56688207953431244549195846153, −9.353947331591675037902049540149, −9.113496501605046025776006305413, −8.164403455178802724518870404503, −7.05688689709595162935919922587, −5.98781185113118965040088796892, −5.52887203661347798664008758489, −4.32642695301213015027287527612, −3.02023207027786292520927255056, −1.55997073430536695350719043871, 0.71874774658346399792727741135, 1.52335944999154391151049063505, 3.48392648742602161550445564652, 4.06793360532312164078620261846, 5.82510295491695036229035186755, 6.73050875171330585748794361335, 7.09268257903118618225755557172, 8.203191424396666187338901476975, 9.021677745890048283889021659533, 9.985919782634462078401151809334

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