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Properties

Label 2-85-17.13-c1-0-4
Degree $2$
Conductor $85$
Sign $-0.582 + 0.813i$
Analytic cond. $0.678728$
Root an. cond. $0.823849$
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  − 1.12i·2-s + (−1.75 − 1.75i)3-s + 0.729·4-s + (−0.707 − 0.707i)5-s + (−1.97 + 1.97i)6-s + (−1.72 + 1.72i)7-s − 3.07i·8-s + 3.16i·9-s + (−0.796 + 0.796i)10-s + (2.57 − 2.57i)11-s + (−1.28 − 1.28i)12-s + 3.64·13-s + (1.94 + 1.94i)14-s + 2.48i·15-s − 2.00·16-s + (3.03 + 2.79i)17-s + ⋯
L(s)  = 1  − 0.796i·2-s + (−1.01 − 1.01i)3-s + 0.364·4-s + (−0.316 − 0.316i)5-s + (−0.807 + 0.807i)6-s + (−0.652 + 0.652i)7-s − 1.08i·8-s + 1.05i·9-s + (−0.252 + 0.252i)10-s + (0.775 − 0.775i)11-s + (−0.369 − 0.369i)12-s + 1.01·13-s + (0.520 + 0.520i)14-s + 0.641i·15-s − 0.502·16-s + (0.735 + 0.677i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.582 + 0.813i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.582 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $-0.582 + 0.813i$
Analytic conductor: \(0.678728\)
Root analytic conductor: \(0.823849\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 85,\ (\ :1/2),\ -0.582 + 0.813i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.356799 - 0.694312i\)
\(L(\frac12)\) \(\approx\) \(0.356799 - 0.694312i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.707 + 0.707i)T \)
17 \( 1 + (-3.03 - 2.79i)T \)
good2 \( 1 + 1.12iT - 2T^{2} \)
3 \( 1 + (1.75 + 1.75i)T + 3iT^{2} \)
7 \( 1 + (1.72 - 1.72i)T - 7iT^{2} \)
11 \( 1 + (-2.57 + 2.57i)T - 11iT^{2} \)
13 \( 1 - 3.64T + 13T^{2} \)
19 \( 1 - 2.61iT - 19T^{2} \)
23 \( 1 + (-0.993 + 0.993i)T - 23iT^{2} \)
29 \( 1 + (-0.601 - 0.601i)T + 29iT^{2} \)
31 \( 1 + (6.67 + 6.67i)T + 31iT^{2} \)
37 \( 1 + (-7.78 - 7.78i)T + 37iT^{2} \)
41 \( 1 + (6.74 - 6.74i)T - 41iT^{2} \)
43 \( 1 - 7.47iT - 43T^{2} \)
47 \( 1 + 5.42T + 47T^{2} \)
53 \( 1 + 12.9iT - 53T^{2} \)
59 \( 1 - 1.40iT - 59T^{2} \)
61 \( 1 + (-0.804 + 0.804i)T - 61iT^{2} \)
67 \( 1 + 2.07T + 67T^{2} \)
71 \( 1 + (-8.69 - 8.69i)T + 71iT^{2} \)
73 \( 1 + (-1.04 - 1.04i)T + 73iT^{2} \)
79 \( 1 + (-6.34 + 6.34i)T - 79iT^{2} \)
83 \( 1 - 2.52iT - 83T^{2} \)
89 \( 1 + 1.66T + 89T^{2} \)
97 \( 1 + (8.67 + 8.67i)T + 97iT^{2} \)
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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

−13.16276311346905077779082464648, −12.64557728806656963092411594339, −11.63173640552095751206473673509, −11.22231695044110640459799201475, −9.704083651183634488097854966337, −8.142032629224278282029845910235, −6.52444304737778355126599652300, −5.94268382243527592318942960408, −3.49252935133358472243338541737, −1.31216762820195459865791169534, 3.76214853329508148560640814744, 5.26265557044903241497367019676, 6.49057523496227188475817216496, 7.36128708413412604624954370803, 9.202076031788220451310121039457, 10.46684105428903158616417567534, 11.19134581102792911903807815159, 12.19760019908486875888439122026, 13.89877614301809732672326780837, 15.03135214327336124766955578524

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