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Properties

Label 2-850-17.13-c1-0-0
Degree $2$
Conductor $850$
Sign $0.405 - 0.913i$
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 − i)3-s − 4-s + (−1 + i)6-s + (−1 + i)7-s + i·8-s − i·9-s + (1 − i)11-s + (1 + i)12-s − 4·13-s + (1 + i)14-s + 16-s + (−4 + i)17-s − 18-s + 6i·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.577 − 0.577i)3-s − 0.5·4-s + (−0.408 + 0.408i)6-s + (−0.377 + 0.377i)7-s + 0.353i·8-s − 0.333i·9-s + (0.301 − 0.301i)11-s + (0.288 + 0.288i)12-s − 1.10·13-s + (0.267 + 0.267i)14-s + 0.250·16-s + (−0.970 + 0.242i)17-s − 0.235·18-s + 1.37i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.405 - 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.405 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(850\)    =    \(2 \cdot 5^{2} \cdot 17\)
Sign: $0.405 - 0.913i$
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: \((2,\ 850,\ (\ :1/2),\ 0.405 - 0.913i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.251785 + 0.163681i\)
\(L(\frac12)\) \(\approx\) \(0.251785 + 0.163681i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4 - i)T \)
good3 \( 1 + (1 + i)T + 3iT^{2} \)
7 \( 1 + (1 - i)T - 7iT^{2} \)
11 \( 1 + (-1 + i)T - 11iT^{2} \)
13 \( 1 + 4T + 13T^{2} \)
19 \( 1 - 6iT - 19T^{2} \)
23 \( 1 + (-1 + i)T - 23iT^{2} \)
29 \( 1 + (-5 - 5i)T + 29iT^{2} \)
31 \( 1 + (3 + 3i)T + 31iT^{2} \)
37 \( 1 + (1 + i)T + 37iT^{2} \)
41 \( 1 + (7 - 7i)T - 41iT^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 - 10iT - 53T^{2} \)
59 \( 1 + 10iT - 59T^{2} \)
61 \( 1 + (-3 + 3i)T - 61iT^{2} \)
67 \( 1 + 2T + 67T^{2} \)
71 \( 1 + (-5 - 5i)T + 71iT^{2} \)
73 \( 1 + (1 + i)T + 73iT^{2} \)
79 \( 1 + (-1 + i)T - 79iT^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (3 + 3i)T + 97iT^{2} \)
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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

−10.42959643531000587799331066822, −9.562098069967305636787872667949, −8.863962655594001141375359044957, −7.81397234097499975778816258110, −6.73186045582400478454790022243, −6.05033441988627885998808264410, −5.02237181532791920460579539478, −3.85966432147394005544553624013, −2.72152769807890181422090518180, −1.43149151267648919781531384378, 0.15941108231748573601747323490, 2.44967724431305110705723415718, 4.02676242104101777238251148840, 4.82356842998427384069299099452, 5.46245686289582610526192905021, 6.83464080540749051828655928034, 7.08467787922968493348096311083, 8.347260256136438014329499186700, 9.259588928562946583411733518759, 9.987964799682993570224645312407

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