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Properties

Label 2-85-17.2-c1-0-3
Degree $2$
Conductor $85$
Sign $-0.357 + 0.933i$
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  + (−1.27 − 1.27i)2-s + (0.635 − 0.263i)3-s + 1.26i·4-s + (−0.382 − 0.923i)5-s + (−1.14 − 0.475i)6-s + (1.66 − 4.01i)7-s + (−0.943 + 0.943i)8-s + (−1.78 + 1.78i)9-s + (−0.691 + 1.66i)10-s + (0.0485 + 0.0200i)11-s + (0.331 + 0.801i)12-s − 3.02i·13-s + (−7.24 + 3.00i)14-s + (−0.486 − 0.486i)15-s + 4.93·16-s + (3.12 + 2.69i)17-s + ⋯
L(s)  = 1  + (−0.902 − 0.902i)2-s + (0.366 − 0.151i)3-s + 0.630i·4-s + (−0.171 − 0.413i)5-s + (−0.468 − 0.194i)6-s + (0.628 − 1.51i)7-s + (−0.333 + 0.333i)8-s + (−0.595 + 0.595i)9-s + (−0.218 + 0.527i)10-s + (0.0146 + 0.00605i)11-s + (0.0958 + 0.231i)12-s − 0.839i·13-s + (−1.93 + 0.801i)14-s + (−0.125 − 0.125i)15-s + 1.23·16-s + (0.757 + 0.653i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.357 + 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.357 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $-0.357 + 0.933i$
Analytic conductor: \(0.678728\)
Root analytic conductor: \(0.823849\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 85,\ (\ :1/2),\ -0.357 + 0.933i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.388946 - 0.565370i\)
\(L(\frac12)\) \(\approx\) \(0.388946 - 0.565370i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.382 + 0.923i)T \)
17 \( 1 + (-3.12 - 2.69i)T \)
good2 \( 1 + (1.27 + 1.27i)T + 2iT^{2} \)
3 \( 1 + (-0.635 + 0.263i)T + (2.12 - 2.12i)T^{2} \)
7 \( 1 + (-1.66 + 4.01i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (-0.0485 - 0.0200i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + 3.02iT - 13T^{2} \)
19 \( 1 + (-5.52 - 5.52i)T + 19iT^{2} \)
23 \( 1 + (-0.962 - 0.398i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (0.161 + 0.388i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (1.27 - 0.529i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (-0.311 + 0.128i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-2.52 + 6.09i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (7.06 - 7.06i)T - 43iT^{2} \)
47 \( 1 - 6.13iT - 47T^{2} \)
53 \( 1 + (8.52 + 8.52i)T + 53iT^{2} \)
59 \( 1 + (-3.60 + 3.60i)T - 59iT^{2} \)
61 \( 1 + (2.28 - 5.51i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 - 0.916T + 67T^{2} \)
71 \( 1 + (-3.86 + 1.59i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (-2.06 - 4.98i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-9.22 - 3.82i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (4.61 + 4.61i)T + 83iT^{2} \)
89 \( 1 - 10.2iT - 89T^{2} \)
97 \( 1 + (7.35 + 17.7i)T + (-68.5 + 68.5i)T^{2} \)
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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

−13.94098428678040777106638408339, −12.67544502985314878999553910183, −11.42477530148288017588409255862, −10.58689250012258831456892689253, −9.731188217355995470481937114084, −8.170356790309723809215911726870, −7.76876071815952544736206924410, −5.39028945694711164175971265103, −3.44413857038071724175409746614, −1.33827655602701050561650597032, 2.99659937610804764127536307363, 5.40151137997810538548560121226, 6.71503010222998048490363436435, 7.948997564671410357385020914539, 8.995598883797923182474165222500, 9.497012631918557886392349026829, 11.46867239653183029437845856424, 12.12540885740275242910751511586, 14.00558851564583653151953233849, 14.97119555941627581075075297323

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