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Properties

Label 2-85-17.2-c1-0-5
Degree $2$
Conductor $85$
Sign $0.925 + 0.377i$
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  + (0.213 + 0.213i)2-s + (0.980 − 0.406i)3-s − 1.90i·4-s + (−0.382 − 0.923i)5-s + (0.295 + 0.122i)6-s + (−0.960 + 2.31i)7-s + (0.833 − 0.833i)8-s + (−1.32 + 1.32i)9-s + (0.115 − 0.278i)10-s + (2.25 + 0.935i)11-s + (−0.775 − 1.87i)12-s + 5.61i·13-s + (−0.699 + 0.289i)14-s + (−0.750 − 0.750i)15-s − 3.46·16-s + (2.76 − 3.06i)17-s + ⋯
L(s)  = 1  + (0.150 + 0.150i)2-s + (0.565 − 0.234i)3-s − 0.954i·4-s + (−0.171 − 0.413i)5-s + (0.120 + 0.0500i)6-s + (−0.363 + 0.876i)7-s + (0.294 − 0.294i)8-s + (−0.441 + 0.441i)9-s + (0.0365 − 0.0881i)10-s + (0.681 + 0.282i)11-s + (−0.223 − 0.540i)12-s + 1.55i·13-s + (−0.186 + 0.0774i)14-s + (−0.193 − 0.193i)15-s − 0.865·16-s + (0.669 − 0.742i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.377i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.925 + 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $0.925 + 0.377i$
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: \((2,\ 85,\ (\ :1/2),\ 0.925 + 0.377i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11238 - 0.218319i\)
\(L(\frac12)\) \(\approx\) \(1.11238 - 0.218319i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.76 + 3.06i)T \)
good2 \( 1 + (-0.213 - 0.213i)T + 2iT^{2} \)
3 \( 1 + (-0.980 + 0.406i)T + (2.12 - 2.12i)T^{2} \)
7 \( 1 + (0.960 - 2.31i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (-2.25 - 0.935i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 - 5.61iT - 13T^{2} \)
19 \( 1 + (5.04 + 5.04i)T + 19iT^{2} \)
23 \( 1 + (-0.795 - 0.329i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (1.43 + 3.46i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (2.07 - 0.860i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (-4.71 + 1.95i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-4.72 + 11.4i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (1.85 - 1.85i)T - 43iT^{2} \)
47 \( 1 - 2.30iT - 47T^{2} \)
53 \( 1 + (-1.96 - 1.96i)T + 53iT^{2} \)
59 \( 1 + (5.26 - 5.26i)T - 59iT^{2} \)
61 \( 1 + (0.346 - 0.837i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + 6.69T + 67T^{2} \)
71 \( 1 + (-0.222 + 0.0921i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (2.47 + 5.96i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-13.5 - 5.62i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (9.82 + 9.82i)T + 83iT^{2} \)
89 \( 1 + 0.395iT - 89T^{2} \)
97 \( 1 + (-3.42 - 8.27i)T + (-68.5 + 68.5i)T^{2} \)
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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

−14.24802959558016495347739827000, −13.38729402654983503156907229571, −12.06357102849648184375988677131, −11.04176385105724015416144635446, −9.311363724320658618405204762104, −8.965573362980959817892945584535, −7.16731934356936293449752271708, −5.90954251001310487501644889958, −4.52284321921065619763693254788, −2.20005399702633186438631002513, 3.18196813149302897793044728956, 3.91157390401259137900895050355, 6.22325757238837303957906756774, 7.71825536198955041337226755347, 8.500528347406063545162550398691, 9.999887615695134279543608455228, 11.05814860759571933107578131959, 12.36674382202784426789171796136, 13.16919810305998613209454708086, 14.39141701119390157032858222471

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