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Properties

Label 2-850-17.13-c1-0-7
Degree $2$
Conductor $850$
Sign $0.122 - 0.992i$
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 + (1.70 + 1.70i)3-s − 4-s + (1.70 − 1.70i)6-s + (−1 + i)7-s + i·8-s + 2.82i·9-s + (−4.41 + 4.41i)11-s + (−1.70 − 1.70i)12-s + 3·13-s + (1 + i)14-s + 16-s + (−3.53 + 2.12i)17-s + 2.82·18-s + 1.24i·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.985 + 0.985i)3-s − 0.5·4-s + (0.696 − 0.696i)6-s + (−0.377 + 0.377i)7-s + 0.353i·8-s + 0.942i·9-s + (−1.33 + 1.33i)11-s + (−0.492 − 0.492i)12-s + 0.832·13-s + (0.267 + 0.267i)14-s + 0.250·16-s + (−0.857 + 0.514i)17-s + 0.666·18-s + 0.285i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.122 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.122 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(850\)    =    \(2 \cdot 5^{2} \cdot 17\)
Sign: $0.122 - 0.992i$
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: \((2,\ 850,\ (\ :1/2),\ 0.122 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16829 + 1.03329i\)
\(L(\frac12)\) \(\approx\) \(1.16829 + 1.03329i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.53 - 2.12i)T \)
good3 \( 1 + (-1.70 - 1.70i)T + 3iT^{2} \)
7 \( 1 + (1 - i)T - 7iT^{2} \)
11 \( 1 + (4.41 - 4.41i)T - 11iT^{2} \)
13 \( 1 - 3T + 13T^{2} \)
19 \( 1 - 1.24iT - 19T^{2} \)
23 \( 1 + (2.82 - 2.82i)T - 23iT^{2} \)
29 \( 1 + (-0.707 - 0.707i)T + 29iT^{2} \)
31 \( 1 + (-7.36 - 7.36i)T + 31iT^{2} \)
37 \( 1 + (3.24 + 3.24i)T + 37iT^{2} \)
41 \( 1 + (-1.58 + 1.58i)T - 41iT^{2} \)
43 \( 1 + 12.2iT - 43T^{2} \)
47 \( 1 - 4.41T + 47T^{2} \)
53 \( 1 - 3iT - 53T^{2} \)
59 \( 1 - 6.89iT - 59T^{2} \)
61 \( 1 + (-1.87 + 1.87i)T - 61iT^{2} \)
67 \( 1 - 2.48T + 67T^{2} \)
71 \( 1 + (-2.29 - 2.29i)T + 71iT^{2} \)
73 \( 1 + (4.36 + 4.36i)T + 73iT^{2} \)
79 \( 1 + (8.24 - 8.24i)T - 79iT^{2} \)
83 \( 1 - 4.24iT - 83T^{2} \)
89 \( 1 - 5.48T + 89T^{2} \)
97 \( 1 + (4.12 + 4.12i)T + 97iT^{2} \)
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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

−10.32385633440884296043644101620, −9.674708386917633464537013254231, −8.802634427448273357752119410690, −8.299119314179656622759427767009, −7.14421767467650115528285186245, −5.73700370047887424424130565433, −4.67167820542052246345106106572, −3.90227813510474451603400544847, −2.91116516467249981356759512570, −2.03227463506447347816024187161, 0.65710634214172662455328054846, 2.46200061442239177351263155582, 3.33267270459464070028522390797, 4.65227444461460451360459633276, 5.97929583843138150803825606952, 6.58999761359573228457737170864, 7.59002070891298547810864378977, 8.256847447933813019895573484881, 8.656077595071778500645043837184, 9.773673771962375245665599334870

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