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Properties

Label 2-850-85.82-c1-0-12
Degree $2$
Conductor $850$
Sign $-0.535 - 0.844i$
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 + (1.49 + 2.23i)3-s + (0.707 + 0.707i)4-s + (0.524 + 2.63i)6-s + (0.501 + 2.52i)7-s + (0.382 + 0.923i)8-s + (−1.61 + 3.89i)9-s + (−2.07 + 0.412i)11-s + (−0.524 + 2.63i)12-s + 0.669·13-s + (−0.501 + 2.52i)14-s + i·16-s + (3.66 − 1.88i)17-s + (−2.97 + 2.97i)18-s + (−2.43 − 5.87i)19-s + ⋯
L(s)  = 1  + (0.653 + 0.270i)2-s + (0.861 + 1.28i)3-s + (0.353 + 0.353i)4-s + (0.213 + 1.07i)6-s + (0.189 + 0.952i)7-s + (0.135 + 0.326i)8-s + (−0.537 + 1.29i)9-s + (−0.625 + 0.124i)11-s + (−0.151 + 0.760i)12-s + 0.185·13-s + (−0.133 + 0.673i)14-s + 0.250i·16-s + (0.889 − 0.455i)17-s + (−0.702 + 0.702i)18-s + (−0.557 − 1.34i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.535 - 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(850\)    =    \(2 \cdot 5^{2} \cdot 17\)
Sign: $-0.535 - 0.844i$
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: \((2,\ 850,\ (\ :1/2),\ -0.535 - 0.844i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42834 + 2.59843i\)
\(L(\frac12)\) \(\approx\) \(1.42834 + 2.59843i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.66 + 1.88i)T \)
good3 \( 1 + (-1.49 - 2.23i)T + (-1.14 + 2.77i)T^{2} \)
7 \( 1 + (-0.501 - 2.52i)T + (-6.46 + 2.67i)T^{2} \)
11 \( 1 + (2.07 - 0.412i)T + (10.1 - 4.20i)T^{2} \)
13 \( 1 - 0.669T + 13T^{2} \)
19 \( 1 + (2.43 + 5.87i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (4.02 + 2.68i)T + (8.80 + 21.2i)T^{2} \)
29 \( 1 + (-1.33 + 0.894i)T + (11.0 - 26.7i)T^{2} \)
31 \( 1 + (-8.76 - 1.74i)T + (28.6 + 11.8i)T^{2} \)
37 \( 1 + (7.79 - 5.20i)T + (14.1 - 34.1i)T^{2} \)
41 \( 1 + (-1.26 - 0.844i)T + (15.6 + 37.8i)T^{2} \)
43 \( 1 + (-6.08 + 2.52i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 - 8.92iT - 47T^{2} \)
53 \( 1 + (-4.10 + 9.91i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (3.21 + 1.33i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (-5.81 + 8.70i)T + (-23.3 - 56.3i)T^{2} \)
67 \( 1 + (7.04 - 7.04i)T - 67iT^{2} \)
71 \( 1 + (1.31 - 6.58i)T + (-65.5 - 27.1i)T^{2} \)
73 \( 1 + (2.39 - 12.0i)T + (-67.4 - 27.9i)T^{2} \)
79 \( 1 + (-1.27 - 6.40i)T + (-72.9 + 30.2i)T^{2} \)
83 \( 1 + (11.2 + 4.65i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (0.324 + 0.324i)T + 89iT^{2} \)
97 \( 1 + (-2.88 + 14.4i)T + (-89.6 - 37.1i)T^{2} \)
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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

−10.28717102847406027432529793228, −9.690246092514426844928423864548, −8.551283212945521138729627171165, −8.341419033779014397826996683977, −7.01656717141563763329987202254, −5.81607119529733911312968378776, −4.94280358973915730396721779192, −4.30131773258330439279429854816, −3.05772072150203068652615696416, −2.46924244037658598213693823765, 1.15074717131965045021278800141, 2.16483409359094465297433092111, 3.33613728887238231395476663185, 4.18411634768745263737432639841, 5.65264211067283952117177061543, 6.44429540371150134642826892660, 7.59397529836967791001760648638, 7.81164462387593341005574612160, 8.826566542968861205157926968887, 10.23567222889493110883784403602

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