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Properties

Label 2-85-17.4-c1-0-0
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: $\chi_{85} (21, \cdot )$
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

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

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