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Properties

Label 2-85-17.2-c1-0-2
Degree $2$
Conductor $85$
Sign $0.729 - 0.683i$
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  + (1.01 + 1.01i)2-s + (−0.101 + 0.0420i)3-s + 0.0689i·4-s + (0.382 + 0.923i)5-s + (−0.146 − 0.0604i)6-s + (−0.265 + 0.642i)7-s + (1.96 − 1.96i)8-s + (−2.11 + 2.11i)9-s + (−0.550 + 1.32i)10-s + (−4.48 − 1.85i)11-s + (−0.00290 − 0.00700i)12-s − 5.63i·13-s + (−0.923 + 0.382i)14-s + (−0.0777 − 0.0777i)15-s + 4.13·16-s + (1.63 + 3.78i)17-s + ⋯
L(s)  = 1  + (0.719 + 0.719i)2-s + (−0.0586 + 0.0242i)3-s + 0.0344i·4-s + (0.171 + 0.413i)5-s + (−0.0596 − 0.0246i)6-s + (−0.100 + 0.242i)7-s + (0.694 − 0.694i)8-s + (−0.704 + 0.704i)9-s + (−0.174 + 0.420i)10-s + (−1.35 − 0.559i)11-s + (−0.000837 − 0.00202i)12-s − 1.56i·13-s + (−0.246 + 0.102i)14-s + (−0.0200 − 0.0200i)15-s + 1.03·16-s + (0.395 + 0.918i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 - 0.683i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.729 - 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $0.729 - 0.683i$
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: \((2,\ 85,\ (\ :1/2),\ 0.729 - 0.683i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19127 + 0.470911i\)
\(L(\frac12)\) \(\approx\) \(1.19127 + 0.470911i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.63 - 3.78i)T \)
good2 \( 1 + (-1.01 - 1.01i)T + 2iT^{2} \)
3 \( 1 + (0.101 - 0.0420i)T + (2.12 - 2.12i)T^{2} \)
7 \( 1 + (0.265 - 0.642i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (4.48 + 1.85i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + 5.63iT - 13T^{2} \)
19 \( 1 + (1.64 + 1.64i)T + 19iT^{2} \)
23 \( 1 + (-4.28 - 1.77i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (-2.48 - 6.01i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (6.12 - 2.53i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (-0.109 + 0.0453i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (0.412 - 0.996i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (0.453 - 0.453i)T - 43iT^{2} \)
47 \( 1 + 4.93iT - 47T^{2} \)
53 \( 1 + (-8.47 - 8.47i)T + 53iT^{2} \)
59 \( 1 + (-7.01 + 7.01i)T - 59iT^{2} \)
61 \( 1 + (-0.613 + 1.48i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 - 2.99T + 67T^{2} \)
71 \( 1 + (4.33 - 1.79i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (2.10 + 5.08i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (13.7 + 5.68i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (3.56 + 3.56i)T + 83iT^{2} \)
89 \( 1 + 2.35iT - 89T^{2} \)
97 \( 1 + (1.03 + 2.49i)T + (-68.5 + 68.5i)T^{2} \)
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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

−14.50919908312216750792348459134, −13.37344708825482249993339512437, −12.78555265985954642496773408119, −10.80184772454441940688191000553, −10.41389371311337720006748720023, −8.450762980282738385565514630630, −7.34897298848361678039903604460, −5.76506917375159869962259661269, −5.25186822424864276388264760609, −3.05967928487064226286831040859, 2.49820052709460728138734361082, 4.15843029420639913947254401991, 5.39953461395510468291446127521, 7.16980342576747190446254547334, 8.596018225201871138955172673012, 9.895528340276405978589085530451, 11.24789448963188756117000756444, 12.04486270958082182843276153501, 12.98569773671864833455051182483, 13.84400169206765022811673230026

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