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Properties

Label 2-850-17.13-c1-0-26
Degree $2$
Conductor $850$
Sign $-0.904 - 0.426i$
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  − i·2-s + (−1.90 − 1.90i)3-s − 4-s + (−1.90 + 1.90i)6-s + (2.69 − 2.69i)7-s + i·8-s + 4.28i·9-s + (3.82 − 3.82i)11-s + (1.90 + 1.90i)12-s − 2.11·13-s + (−2.69 − 2.69i)14-s + 16-s + (−0.908 − 4.02i)17-s + 4.28·18-s − 7.10i·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−1.10 − 1.10i)3-s − 0.5·4-s + (−0.779 + 0.779i)6-s + (1.02 − 1.02i)7-s + 0.353i·8-s + 1.42i·9-s + (1.15 − 1.15i)11-s + (0.550 + 0.550i)12-s − 0.586·13-s + (−0.721 − 0.721i)14-s + 0.250·16-s + (−0.220 − 0.975i)17-s + 1.00·18-s − 1.62i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 - 0.426i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.904 - 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(850\)    =    \(2 \cdot 5^{2} \cdot 17\)
Sign: $-0.904 - 0.426i$
Analytic conductor: \(6.78728\)
Root analytic conductor: \(2.60524\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{850} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 850,\ (\ :1/2),\ -0.904 - 0.426i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.222703 + 0.994178i\)
\(L(\frac12)\) \(\approx\) \(0.222703 + 0.994178i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 \)
17 \( 1 + (0.908 + 4.02i)T \)
good3 \( 1 + (1.90 + 1.90i)T + 3iT^{2} \)
7 \( 1 + (-2.69 + 2.69i)T - 7iT^{2} \)
11 \( 1 + (-3.82 + 3.82i)T - 11iT^{2} \)
13 \( 1 + 2.11T + 13T^{2} \)
19 \( 1 + 7.10iT - 19T^{2} \)
23 \( 1 + (-0.585 + 0.585i)T - 23iT^{2} \)
29 \( 1 + (-3.49 - 3.49i)T + 29iT^{2} \)
31 \( 1 + (0.779 + 0.779i)T + 31iT^{2} \)
37 \( 1 + (-7.52 - 7.52i)T + 37iT^{2} \)
41 \( 1 + (4.39 - 4.39i)T - 41iT^{2} \)
43 \( 1 - 7.81iT - 43T^{2} \)
47 \( 1 + 2.29T + 47T^{2} \)
53 \( 1 + 3.28iT - 53T^{2} \)
59 \( 1 + 0.555iT - 59T^{2} \)
61 \( 1 + (-2.89 + 2.89i)T - 61iT^{2} \)
67 \( 1 - 3.97T + 67T^{2} \)
71 \( 1 + (4.59 + 4.59i)T + 71iT^{2} \)
73 \( 1 + (7.45 + 7.45i)T + 73iT^{2} \)
79 \( 1 + (9.61 - 9.61i)T - 79iT^{2} \)
83 \( 1 - 4.04iT - 83T^{2} \)
89 \( 1 + 11.5T + 89T^{2} \)
97 \( 1 + (-0.600 - 0.600i)T + 97iT^{2} \)
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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

−9.920293666893193757359941877377, −8.860400436596583012811897533497, −7.897850678906681521414739950837, −6.99585262807052972866382013324, −6.38950873029964362860951459612, −5.07869111605367247375296072039, −4.48581724619899995667787966151, −2.92481290835022831738068855657, −1.37340181775045590364771481408, −0.65309757836195555815646416128, 1.83912353696628495215158021427, 4.01301594205041649020233746286, 4.50831386080924658383799506347, 5.53249250567467896633882862744, 5.97401305183480450728319597127, 7.10627650275090400400137261747, 8.200226014956690316310789830313, 9.047496512101520098293021208522, 9.868083128251760000144692833474, 10.47280411247258515802795753731

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