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Properties

Label 2-850-85.23-c1-0-19
Degree $2$
Conductor $850$
Sign $-0.281 + 0.959i$
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 + (0.628 + 0.419i)3-s + (0.707 − 0.707i)4-s + (−0.741 − 0.147i)6-s + (−1.27 − 0.252i)7-s + (−0.382 + 0.923i)8-s + (−0.929 − 2.24i)9-s + (−0.546 + 2.74i)11-s + (0.741 − 0.147i)12-s − 4.10·13-s + (1.27 − 0.252i)14-s − i·16-s + (2.50 − 3.27i)17-s + (1.71 + 1.71i)18-s + (1.17 − 2.83i)19-s + ⋯
L(s)  = 1  + (−0.653 + 0.270i)2-s + (0.362 + 0.242i)3-s + (0.353 − 0.353i)4-s + (−0.302 − 0.0601i)6-s + (−0.480 − 0.0955i)7-s + (−0.135 + 0.326i)8-s + (−0.309 − 0.747i)9-s + (−0.164 + 0.827i)11-s + (0.213 − 0.0425i)12-s − 1.13·13-s + (0.339 − 0.0675i)14-s − 0.250i·16-s + (0.607 − 0.794i)17-s + (0.404 + 0.404i)18-s + (0.269 − 0.650i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.281 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.281 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(850\)    =    \(2 \cdot 5^{2} \cdot 17\)
Sign: $-0.281 + 0.959i$
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: \((2,\ 850,\ (\ :1/2),\ -0.281 + 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.319472 - 0.426693i\)
\(L(\frac12)\) \(\approx\) \(0.319472 - 0.426693i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.50 + 3.27i)T \)
good3 \( 1 + (-0.628 - 0.419i)T + (1.14 + 2.77i)T^{2} \)
7 \( 1 + (1.27 + 0.252i)T + (6.46 + 2.67i)T^{2} \)
11 \( 1 + (0.546 - 2.74i)T + (-10.1 - 4.20i)T^{2} \)
13 \( 1 + 4.10T + 13T^{2} \)
19 \( 1 + (-1.17 + 2.83i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (4.19 + 6.28i)T + (-8.80 + 21.2i)T^{2} \)
29 \( 1 + (3.51 - 5.26i)T + (-11.0 - 26.7i)T^{2} \)
31 \( 1 + (0.355 + 1.78i)T + (-28.6 + 11.8i)T^{2} \)
37 \( 1 + (0.190 - 0.284i)T + (-14.1 - 34.1i)T^{2} \)
41 \( 1 + (1.73 + 2.59i)T + (-15.6 + 37.8i)T^{2} \)
43 \( 1 + (7.35 + 3.04i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 + 7.90iT - 47T^{2} \)
53 \( 1 + (-4.29 - 10.3i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-9.44 + 3.91i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-5.15 + 3.44i)T + (23.3 - 56.3i)T^{2} \)
67 \( 1 + (0.944 + 0.944i)T + 67iT^{2} \)
71 \( 1 + (7.94 - 1.58i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (1.75 - 0.349i)T + (67.4 - 27.9i)T^{2} \)
79 \( 1 + (16.2 + 3.23i)T + (72.9 + 30.2i)T^{2} \)
83 \( 1 + (-3.31 + 1.37i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-7.01 + 7.01i)T - 89iT^{2} \)
97 \( 1 + (8.19 - 1.63i)T + (89.6 - 37.1i)T^{2} \)
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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

−9.893253682732586701497368943587, −9.171688406773251078255812981173, −8.386675501295990192293901100676, −7.25975263092855914486360377308, −6.83461752961745385717865530140, −5.60349489244370163774014540937, −4.59914267834896282871534134212, −3.27456221313210193703170769686, −2.24578239173049638906999663725, −0.28871482091835882311999453139, 1.68429561528171041505709818085, 2.81105566894806176927570429018, 3.73892693970953727838659584573, 5.30936889095665157263751183274, 6.12290161823223649146936682143, 7.39783660627566440304490188810, 7.952612767480331888778209347813, 8.644956482135533923505753724546, 9.825077510270002715582173028513, 10.08672796266128276833229751383

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