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Properties

Label 2-85-85.9-c1-0-2
Degree $2$
Conductor $85$
Sign $0.992 + 0.122i$
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  + (0.352 + 0.352i)2-s + (0.0655 − 0.158i)3-s − 1.75i·4-s + (2.22 − 0.233i)5-s + (0.0789 − 0.0327i)6-s + (−3.57 + 1.48i)7-s + (1.32 − 1.32i)8-s + (2.10 + 2.10i)9-s + (0.867 + 0.702i)10-s + (−2.71 + 1.12i)11-s + (−0.277 − 0.114i)12-s + 0.983·13-s + (−1.78 − 0.739i)14-s + (0.108 − 0.367i)15-s − 2.56·16-s + (−3.97 − 1.11i)17-s + ⋯
L(s)  = 1  + (0.249 + 0.249i)2-s + (0.0378 − 0.0913i)3-s − 0.875i·4-s + (0.994 − 0.104i)5-s + (0.0322 − 0.0133i)6-s + (−1.35 + 0.560i)7-s + (0.468 − 0.468i)8-s + (0.700 + 0.700i)9-s + (0.274 + 0.222i)10-s + (−0.817 + 0.338i)11-s + (−0.0799 − 0.0331i)12-s + 0.272·13-s + (−0.477 − 0.197i)14-s + (0.0281 − 0.0948i)15-s − 0.641·16-s + (−0.962 − 0.269i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.122i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $0.992 + 0.122i$
Analytic conductor: \(0.678728\)
Root analytic conductor: \(0.823849\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 85,\ (\ :1/2),\ 0.992 + 0.122i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12583 - 0.0689475i\)
\(L(\frac12)\) \(\approx\) \(1.12583 - 0.0689475i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.22 + 0.233i)T \)
17 \( 1 + (3.97 + 1.11i)T \)
good2 \( 1 + (-0.352 - 0.352i)T + 2iT^{2} \)
3 \( 1 + (-0.0655 + 0.158i)T + (-2.12 - 2.12i)T^{2} \)
7 \( 1 + (3.57 - 1.48i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (2.71 - 1.12i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 - 0.983T + 13T^{2} \)
19 \( 1 + (1.81 - 1.81i)T - 19iT^{2} \)
23 \( 1 + (1.15 + 2.77i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (-0.210 + 0.507i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (-6.98 - 2.89i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (3.76 - 9.09i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (2.83 + 6.84i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (-5.85 + 5.85i)T - 43iT^{2} \)
47 \( 1 - 10.9T + 47T^{2} \)
53 \( 1 + (2.53 + 2.53i)T + 53iT^{2} \)
59 \( 1 + (-0.216 - 0.216i)T + 59iT^{2} \)
61 \( 1 + (2.60 + 6.28i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + 5.09iT - 67T^{2} \)
71 \( 1 + (3.33 + 1.38i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (-4.62 - 1.91i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (11.4 - 4.76i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (5.74 + 5.74i)T + 83iT^{2} \)
89 \( 1 + 13.2iT - 89T^{2} \)
97 \( 1 + (-4.18 - 1.73i)T + (68.5 + 68.5i)T^{2} \)
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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

−13.97833987045927277548935689243, −13.33072018912523887053738152390, −12.52350576580854581932063491295, −10.47096676235807926491150419323, −10.05710824637119186164645388167, −8.859276416804748016895937914233, −6.88672146216983981875664605944, −6.01793249832006547079345655237, −4.80572087540721883376827377287, −2.28553238167153591848681809117, 2.80434187622316532833073440321, 4.17277877480860633602450274244, 6.16517849060474482741770615816, 7.20526374946343897246947009936, 8.875250355760475775455998716885, 9.918971402220433003942939858254, 10.93048220805254405152833006369, 12.54889627371734169776664127169, 13.16734805802509375354958571629, 13.76370139201821994115295430954

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