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Properties

Label 2-85-17.2-c1-0-1
Degree $2$
Conductor $85$
Sign $0.0229 - 0.999i$
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  + (1.66 + 1.66i)2-s + (−1.44 + 0.600i)3-s + 3.53i·4-s + (−0.382 − 0.923i)5-s + (−3.41 − 1.41i)6-s + (1.37 − 3.30i)7-s + (−2.54 + 2.54i)8-s + (−0.379 + 0.379i)9-s + (0.900 − 2.17i)10-s + (2.29 + 0.950i)11-s + (−2.12 − 5.12i)12-s + 1.25i·13-s + (7.78 − 3.22i)14-s + (1.10 + 1.10i)15-s − 1.40·16-s + (−3.84 − 1.48i)17-s + ⋯
L(s)  = 1  + (1.17 + 1.17i)2-s + (−0.837 + 0.346i)3-s + 1.76i·4-s + (−0.171 − 0.413i)5-s + (−1.39 − 0.576i)6-s + (0.518 − 1.25i)7-s + (−0.900 + 0.900i)8-s + (−0.126 + 0.126i)9-s + (0.284 − 0.687i)10-s + (0.691 + 0.286i)11-s + (−0.612 − 1.47i)12-s + 0.349i·13-s + (2.08 − 0.861i)14-s + (0.286 + 0.286i)15-s − 0.352·16-s + (−0.932 − 0.360i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0229 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0229 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $0.0229 - 0.999i$
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: \((2,\ 85,\ (\ :1/2),\ 0.0229 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.942727 + 0.921305i\)
\(L(\frac12)\) \(\approx\) \(0.942727 + 0.921305i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.84 + 1.48i)T \)
good2 \( 1 + (-1.66 - 1.66i)T + 2iT^{2} \)
3 \( 1 + (1.44 - 0.600i)T + (2.12 - 2.12i)T^{2} \)
7 \( 1 + (-1.37 + 3.30i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (-2.29 - 0.950i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 - 1.25iT - 13T^{2} \)
19 \( 1 + (2.56 + 2.56i)T + 19iT^{2} \)
23 \( 1 + (8.16 + 3.38i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (-3.35 - 8.09i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (-2.25 + 0.935i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (4.25 - 1.76i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (1.65 - 3.99i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (-5.33 + 5.33i)T - 43iT^{2} \)
47 \( 1 - 11.3iT - 47T^{2} \)
53 \( 1 + (-4.02 - 4.02i)T + 53iT^{2} \)
59 \( 1 + (-3.16 + 3.16i)T - 59iT^{2} \)
61 \( 1 + (-0.0929 + 0.224i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 - 7.23T + 67T^{2} \)
71 \( 1 + (1.69 - 0.703i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (2.09 + 5.05i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (8.34 + 3.45i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (5.17 + 5.17i)T + 83iT^{2} \)
89 \( 1 + 2.19iT - 89T^{2} \)
97 \( 1 + (3.75 + 9.07i)T + (-68.5 + 68.5i)T^{2} \)
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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

−14.30496439312936922923551874724, −13.84983550318294212361873078259, −12.58488375060787340484879144650, −11.53687848323682598532315697526, −10.42280776804998650585594513220, −8.500946322437058259752560113811, −7.18741050369278288064398588368, −6.22731609589481231464369395474, −4.72905802468583350251885751116, −4.26034397746175427542557155407, 2.18474600623475618755788602905, 3.94720409311614932351643910379, 5.54531918365014681491602552639, 6.27419058980108650212787108420, 8.471431824616060161069632028536, 10.16759518691375815526656414753, 11.39234666623152878245090656829, 11.80749724945863707314662558093, 12.51718764421538390422779266744, 13.80258182727891236922187801650

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