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Properties

Label 2-85-17.15-c1-0-2
Degree $2$
Conductor $85$
Sign $0.719 + 0.693i$
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  + (−0.254 − 0.254i)2-s + (0.0207 + 0.0501i)3-s − 1.87i·4-s + (0.923 − 0.382i)5-s + (0.00747 − 0.0180i)6-s + (0.275 + 0.114i)7-s + (−0.985 + 0.985i)8-s + (2.11 − 2.11i)9-s + (−0.332 − 0.137i)10-s + (−1.05 + 2.55i)11-s + (0.0937 − 0.0388i)12-s + 1.97i·13-s + (−0.0411 − 0.0994i)14-s + (0.0383 + 0.0383i)15-s − 3.23·16-s + (−1.21 + 3.94i)17-s + ⋯
L(s)  = 1  + (−0.180 − 0.180i)2-s + (0.0119 + 0.0289i)3-s − 0.935i·4-s + (0.413 − 0.171i)5-s + (0.00305 − 0.00736i)6-s + (0.104 + 0.0432i)7-s + (−0.348 + 0.348i)8-s + (0.706 − 0.706i)9-s + (−0.105 − 0.0435i)10-s + (−0.319 + 0.770i)11-s + (0.0270 − 0.0112i)12-s + 0.549i·13-s + (−0.0110 − 0.0265i)14-s + (0.00990 + 0.00990i)15-s − 0.809·16-s + (−0.293 + 0.955i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.719 + 0.693i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.719 + 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $0.719 + 0.693i$
Analytic conductor: \(0.678728\)
Root analytic conductor: \(0.823849\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (66, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 85,\ (\ :1/2),\ 0.719 + 0.693i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.870444 - 0.351216i\)
\(L(\frac12)\) \(\approx\) \(0.870444 - 0.351216i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.923 + 0.382i)T \)
17 \( 1 + (1.21 - 3.94i)T \)
good2 \( 1 + (0.254 + 0.254i)T + 2iT^{2} \)
3 \( 1 + (-0.0207 - 0.0501i)T + (-2.12 + 2.12i)T^{2} \)
7 \( 1 + (-0.275 - 0.114i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (1.05 - 2.55i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 - 1.97iT - 13T^{2} \)
19 \( 1 + (1.99 + 1.99i)T + 19iT^{2} \)
23 \( 1 + (2.57 - 6.22i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-4.36 + 1.80i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (-1.15 - 2.79i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (3.60 + 8.70i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-2.87 - 1.19i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (-5.78 + 5.78i)T - 43iT^{2} \)
47 \( 1 + 1.08iT - 47T^{2} \)
53 \( 1 + (1.89 + 1.89i)T + 53iT^{2} \)
59 \( 1 + (6.47 - 6.47i)T - 59iT^{2} \)
61 \( 1 + (-10.3 - 4.28i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + 12.5T + 67T^{2} \)
71 \( 1 + (2.25 + 5.43i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (0.200 - 0.0829i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (4.07 - 9.84i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (11.0 + 11.0i)T + 83iT^{2} \)
89 \( 1 - 1.55iT - 89T^{2} \)
97 \( 1 + (-8.28 + 3.43i)T + (68.5 - 68.5i)T^{2} \)
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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

−14.20750901806910801099956165882, −13.06498781672242605056937784290, −11.95540131090094637377917104131, −10.61440893111850320621373391270, −9.785330268432343865274720369674, −8.861904580878236399163892618880, −7.03817209901263431063813970437, −5.84655394083641571692404764062, −4.38417632166226787253342975634, −1.82640033083294669724485013956, 2.78746953348926650965759202979, 4.60749404670277446830063327538, 6.38052059334660242969843926709, 7.67670222979078573743822863297, 8.558890232574302589390526954754, 10.00744811108340410341144853446, 11.08228881131254135144852388977, 12.42503211866498636853312741959, 13.27674694546926300868903832728, 14.19439879426464434649359606771

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