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Properties

Label 2-85-85.59-c1-0-4
Degree $2$
Conductor $85$
Sign $0.588 + 0.808i$
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  + (0.176 + 0.176i)2-s + (−1.51 − 0.629i)3-s − 1.93i·4-s + (1.62 − 1.53i)5-s + (−0.156 − 0.377i)6-s + (0.547 + 1.32i)7-s + (0.693 − 0.693i)8-s + (−0.210 − 0.210i)9-s + (0.556 + 0.0165i)10-s + (1.03 + 2.48i)11-s + (−1.21 + 2.94i)12-s + 0.174·13-s + (−0.136 + 0.329i)14-s + (−3.43 + 1.30i)15-s − 3.63·16-s + (2.27 + 3.44i)17-s + ⋯
L(s)  = 1  + (0.124 + 0.124i)2-s + (−0.876 − 0.363i)3-s − 0.969i·4-s + (0.727 − 0.685i)5-s + (−0.0639 − 0.154i)6-s + (0.206 + 0.499i)7-s + (0.245 − 0.245i)8-s + (−0.0703 − 0.0703i)9-s + (0.175 + 0.00523i)10-s + (0.310 + 0.750i)11-s + (−0.351 + 0.849i)12-s + 0.0484·13-s + (−0.0364 + 0.0879i)14-s + (−0.887 + 0.336i)15-s − 0.908·16-s + (0.550 + 0.834i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.588 + 0.808i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.588 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $0.588 + 0.808i$
Analytic conductor: \(0.678728\)
Root analytic conductor: \(0.823849\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 85,\ (\ :1/2),\ 0.588 + 0.808i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.790864 - 0.402276i\)
\(L(\frac12)\) \(\approx\) \(0.790864 - 0.402276i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.62 + 1.53i)T \)
17 \( 1 + (-2.27 - 3.44i)T \)
good2 \( 1 + (-0.176 - 0.176i)T + 2iT^{2} \)
3 \( 1 + (1.51 + 0.629i)T + (2.12 + 2.12i)T^{2} \)
7 \( 1 + (-0.547 - 1.32i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (-1.03 - 2.48i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 - 0.174T + 13T^{2} \)
19 \( 1 + (2.69 - 2.69i)T - 19iT^{2} \)
23 \( 1 + (-2.54 + 1.05i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (-5.94 - 2.46i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (0.188 - 0.454i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (5.30 + 2.19i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-5.54 + 2.29i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (8.00 - 8.00i)T - 43iT^{2} \)
47 \( 1 - 2.65T + 47T^{2} \)
53 \( 1 + (8.73 + 8.73i)T + 53iT^{2} \)
59 \( 1 + (5.72 + 5.72i)T + 59iT^{2} \)
61 \( 1 + (6.41 - 2.65i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + 12.3iT - 67T^{2} \)
71 \( 1 + (4.67 - 11.2i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (5.99 - 14.4i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (4.94 + 11.9i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (-6.06 - 6.06i)T + 83iT^{2} \)
89 \( 1 - 11.4iT - 89T^{2} \)
97 \( 1 + (0.879 - 2.12i)T + (-68.5 - 68.5i)T^{2} \)
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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

−14.22878229174189234290650152262, −12.81305143459083596544905245985, −12.14115272550050885366180279703, −10.80370975621283226437015863879, −9.794941517071743685337967770981, −8.630278816642640461115755766497, −6.62103284979673782865944163306, −5.80528994964535584891758735969, −4.83020201582188431581469610648, −1.55870074942881671037913755154, 2.99467006010778839180334156545, 4.72000607096478589500387763141, 6.16469478457576260365743193729, 7.40084440872480146159075586997, 8.885961996383748209899652919131, 10.40201842860853381902669063420, 11.16797588666866726449340424260, 12.03603254148827562701427630983, 13.51263622640938331255135260043, 14.05738940719631618012341036375

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