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Properties

Label 2-850-85.37-c1-0-22
Degree $2$
Conductor $850$
Sign $0.156 + 0.987i$
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 + (1.98 − 1.32i)3-s + (0.707 + 0.707i)4-s + (−2.34 + 0.465i)6-s + (4.88 − 0.972i)7-s + (−0.382 − 0.923i)8-s + (1.03 − 2.49i)9-s + (−0.514 − 2.58i)11-s + (2.34 + 0.465i)12-s − 3.55·13-s + (−4.88 − 0.972i)14-s + i·16-s + (3.56 − 2.06i)17-s + (−1.91 + 1.91i)18-s + (−1.27 − 3.08i)19-s + ⋯
L(s)  = 1  + (−0.653 − 0.270i)2-s + (1.14 − 0.766i)3-s + (0.353 + 0.353i)4-s + (−0.956 + 0.190i)6-s + (1.84 − 0.367i)7-s + (−0.135 − 0.326i)8-s + (0.345 − 0.832i)9-s + (−0.155 − 0.779i)11-s + (0.676 + 0.134i)12-s − 0.986·13-s + (−1.30 − 0.259i)14-s + 0.250i·16-s + (0.865 − 0.500i)17-s + (−0.450 + 0.450i)18-s + (−0.293 − 0.707i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(850\)    =    \(2 \cdot 5^{2} \cdot 17\)
Sign: $0.156 + 0.987i$
Analytic conductor: \(6.78728\)
Root analytic conductor: \(2.60524\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{850} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 850,\ (\ :1/2),\ 0.156 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48774 - 1.27046i\)
\(L(\frac12)\) \(\approx\) \(1.48774 - 1.27046i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.56 + 2.06i)T \)
good3 \( 1 + (-1.98 + 1.32i)T + (1.14 - 2.77i)T^{2} \)
7 \( 1 + (-4.88 + 0.972i)T + (6.46 - 2.67i)T^{2} \)
11 \( 1 + (0.514 + 2.58i)T + (-10.1 + 4.20i)T^{2} \)
13 \( 1 + 3.55T + 13T^{2} \)
19 \( 1 + (1.27 + 3.08i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (3.96 - 5.93i)T + (-8.80 - 21.2i)T^{2} \)
29 \( 1 + (-0.593 - 0.888i)T + (-11.0 + 26.7i)T^{2} \)
31 \( 1 + (-0.489 + 2.46i)T + (-28.6 - 11.8i)T^{2} \)
37 \( 1 + (-4.15 - 6.21i)T + (-14.1 + 34.1i)T^{2} \)
41 \( 1 + (1.27 - 1.90i)T + (-15.6 - 37.8i)T^{2} \)
43 \( 1 + (-7.30 + 3.02i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 7.15iT - 47T^{2} \)
53 \( 1 + (2.62 - 6.34i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (9.79 + 4.05i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (-5.08 - 3.39i)T + (23.3 + 56.3i)T^{2} \)
67 \( 1 + (8.80 - 8.80i)T - 67iT^{2} \)
71 \( 1 + (14.0 + 2.80i)T + (65.5 + 27.1i)T^{2} \)
73 \( 1 + (-7.24 - 1.44i)T + (67.4 + 27.9i)T^{2} \)
79 \( 1 + (5.45 - 1.08i)T + (72.9 - 30.2i)T^{2} \)
83 \( 1 + (-3.02 - 1.25i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (4.91 + 4.91i)T + 89iT^{2} \)
97 \( 1 + (-13.4 - 2.67i)T + (89.6 + 37.1i)T^{2} \)
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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

−9.855309289604606527296505959962, −8.928000714967611564969027247025, −8.191715259428608411689385632193, −7.67382136761168430330081317396, −7.19496171000096475586070817956, −5.60084658438281316664497645488, −4.46673926187059547533094104407, −3.09462401186672588948920841409, −2.14017662505684706364471675671, −1.15691005305367420519962964803, 1.77783014457140798390379649022, 2.60320206593345019673141227393, 4.20324097219625776917288777020, 4.84344157594028766636887834073, 5.98780917953767037901170982773, 7.59408308928814992570692069078, 7.903715837686957013441616418322, 8.632364319863303315150134455965, 9.424893999222770160620409381284, 10.21833813789103204942483475117

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