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Properties

Label 2-85-17.13-c1-0-5
Degree $2$
Conductor $85$
Sign $-0.952 + 0.304i$
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  − 2.51i·2-s + (−0.887 − 0.887i)3-s − 4.31·4-s + (0.707 + 0.707i)5-s + (−2.22 + 2.22i)6-s + (1.14 − 1.14i)7-s + 5.80i·8-s − 1.42i·9-s + (1.77 − 1.77i)10-s + (−2.32 + 2.32i)11-s + (3.82 + 3.82i)12-s + 6.35·13-s + (−2.86 − 2.86i)14-s − 1.25i·15-s + 5.96·16-s + (−0.768 − 4.05i)17-s + ⋯
L(s)  = 1  − 1.77i·2-s + (−0.512 − 0.512i)3-s − 2.15·4-s + (0.316 + 0.316i)5-s + (−0.909 + 0.909i)6-s + (0.431 − 0.431i)7-s + 2.05i·8-s − 0.475i·9-s + (0.561 − 0.561i)10-s + (−0.700 + 0.700i)11-s + (1.10 + 1.10i)12-s + 1.76·13-s + (−0.766 − 0.766i)14-s − 0.323i·15-s + 1.49·16-s + (−0.186 − 0.982i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 + 0.304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.952 + 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $-0.952 + 0.304i$
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: \((2,\ 85,\ (\ :1/2),\ -0.952 + 0.304i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.124877 - 0.799401i\)
\(L(\frac12)\) \(\approx\) \(0.124877 - 0.799401i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.768 + 4.05i)T \)
good2 \( 1 + 2.51iT - 2T^{2} \)
3 \( 1 + (0.887 + 0.887i)T + 3iT^{2} \)
7 \( 1 + (-1.14 + 1.14i)T - 7iT^{2} \)
11 \( 1 + (2.32 - 2.32i)T - 11iT^{2} \)
13 \( 1 - 6.35T + 13T^{2} \)
19 \( 1 - 0.747iT - 19T^{2} \)
23 \( 1 + (-0.101 + 0.101i)T - 23iT^{2} \)
29 \( 1 + (-6.22 - 6.22i)T + 29iT^{2} \)
31 \( 1 + (-5.10 - 5.10i)T + 31iT^{2} \)
37 \( 1 + (0.439 + 0.439i)T + 37iT^{2} \)
41 \( 1 + (4.49 - 4.49i)T - 41iT^{2} \)
43 \( 1 + 2.74iT - 43T^{2} \)
47 \( 1 + 11.9T + 47T^{2} \)
53 \( 1 + 2.71iT - 53T^{2} \)
59 \( 1 + 12.6iT - 59T^{2} \)
61 \( 1 + (-0.328 + 0.328i)T - 61iT^{2} \)
67 \( 1 - 1.81T + 67T^{2} \)
71 \( 1 + (3.01 + 3.01i)T + 71iT^{2} \)
73 \( 1 + (0.856 + 0.856i)T + 73iT^{2} \)
79 \( 1 + (3.57 - 3.57i)T - 79iT^{2} \)
83 \( 1 - 3.58iT - 83T^{2} \)
89 \( 1 + 10.5T + 89T^{2} \)
97 \( 1 + (4.92 + 4.92i)T + 97iT^{2} \)
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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

−13.37438224957663163590749953527, −12.51727732887984269458626764163, −11.51112950842836935700911668658, −10.78221570045610602864840604547, −9.811487014441597487893363119817, −8.484574285480603727748938226460, −6.67603997415276975541573465569, −4.89271477085151024316889919946, −3.25361316616112751865260897916, −1.36563827234338205544906405839, 4.40170198694495347286724613915, 5.59709554769783033229486120873, 6.23710216114107452558859306894, 8.149157095639185608129496376654, 8.559976702924126159326895352796, 10.16259663421182381543220163336, 11.36778488857008855211238513064, 13.28839778436496098482745670303, 13.71895947333878974007550448317, 15.17074458995310432444567510042

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