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Properties

Label 2-850-17.13-c1-0-17
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 + (0.292 + 0.292i)3-s − 4-s + (0.292 − 0.292i)6-s + (−1 + i)7-s + i·8-s − 2.82i·9-s + (−1.58 + 1.58i)11-s + (−0.292 − 0.292i)12-s + 3·13-s + (1 + i)14-s + 16-s + (3.53 − 2.12i)17-s − 2.82·18-s − 7.24i·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.169 + 0.169i)3-s − 0.5·4-s + (0.119 − 0.119i)6-s + (−0.377 + 0.377i)7-s + 0.353i·8-s − 0.942i·9-s + (−0.478 + 0.478i)11-s + (−0.0845 − 0.0845i)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 − 1.66i·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

\(L(1)\) \(\approx\) \(0.922327 - 1.04283i\)
\(L(\frac12)\) \(\approx\) \(0.922327 - 1.04283i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.292 - 0.292i)T + 3iT^{2} \)
7 \( 1 + (1 - i)T - 7iT^{2} \)
11 \( 1 + (1.58 - 1.58i)T - 11iT^{2} \)
13 \( 1 - 3T + 13T^{2} \)
19 \( 1 + 7.24iT - 19T^{2} \)
23 \( 1 + (-2.82 + 2.82i)T - 23iT^{2} \)
29 \( 1 + (0.707 + 0.707i)T + 29iT^{2} \)
31 \( 1 + (5.36 + 5.36i)T + 31iT^{2} \)
37 \( 1 + (-5.24 - 5.24i)T + 37iT^{2} \)
41 \( 1 + (-4.41 + 4.41i)T - 41iT^{2} \)
43 \( 1 + 3.75iT - 43T^{2} \)
47 \( 1 - 1.58T + 47T^{2} \)
53 \( 1 - 3iT - 53T^{2} \)
59 \( 1 + 12.8iT - 59T^{2} \)
61 \( 1 + (-6.12 + 6.12i)T - 61iT^{2} \)
67 \( 1 + 14.4T + 67T^{2} \)
71 \( 1 + (-3.70 - 3.70i)T + 71iT^{2} \)
73 \( 1 + (-8.36 - 8.36i)T + 73iT^{2} \)
79 \( 1 + (-0.242 + 0.242i)T - 79iT^{2} \)
83 \( 1 + 4.24iT - 83T^{2} \)
89 \( 1 + 11.4T + 89T^{2} \)
97 \( 1 + (-0.121 - 0.121i)T + 97iT^{2} \)
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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

−9.792928641929643171930948952500, −9.316035858628484906174804627578, −8.594587430417202139576266044506, −7.44613429980914007088852648104, −6.44291292838065701549130433527, −5.42544197220060500323131489109, −4.36826912728141451748654689284, −3.31605220674106955059622117537, −2.47740990261862954494754564337, −0.73749579370959235806260053374, 1.42945286990051811107075447952, 3.16656000560235816804874695653, 4.08068577399684227193699904427, 5.45141151500490773865980505005, 5.92607948608329026762596753526, 7.14138543116041787564182575874, 7.86983604127261168191530144105, 8.433605096009157564431371012317, 9.478969152381042346448325714617, 10.44400455177865267159835543640

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