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Properties

Label 2-85-17.4-c1-0-3
Degree $2$
Conductor $85$
Sign $0.471 - 0.881i$
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  + 2.07i·2-s + (1.78 − 1.78i)3-s − 2.28·4-s + (−0.707 + 0.707i)5-s + (3.69 + 3.69i)6-s + (−0.260 − 0.260i)7-s − 0.595i·8-s − 3.36i·9-s + (−1.46 − 1.46i)10-s + (−1.76 − 1.76i)11-s + (−4.08 + 4.08i)12-s − 4.68·13-s + (0.540 − 0.540i)14-s + 2.52i·15-s − 3.34·16-s + (3.84 + 1.48i)17-s + ⋯
L(s)  = 1  + 1.46i·2-s + (1.03 − 1.03i)3-s − 1.14·4-s + (−0.316 + 0.316i)5-s + (1.50 + 1.50i)6-s + (−0.0986 − 0.0986i)7-s − 0.210i·8-s − 1.12i·9-s + (−0.463 − 0.463i)10-s + (−0.532 − 0.532i)11-s + (−1.17 + 1.17i)12-s − 1.30·13-s + (0.144 − 0.144i)14-s + 0.651i·15-s − 0.835·16-s + (0.932 + 0.360i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $0.471 - 0.881i$
Analytic conductor: \(0.678728\)
Root analytic conductor: \(0.823849\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 85,\ (\ :1/2),\ 0.471 - 0.881i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.987545 + 0.591503i\)
\(L(\frac12)\) \(\approx\) \(0.987545 + 0.591503i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 + (-3.84 - 1.48i)T \)
good2 \( 1 - 2.07iT - 2T^{2} \)
3 \( 1 + (-1.78 + 1.78i)T - 3iT^{2} \)
7 \( 1 + (0.260 + 0.260i)T + 7iT^{2} \)
11 \( 1 + (1.76 + 1.76i)T + 11iT^{2} \)
13 \( 1 + 4.68T + 13T^{2} \)
19 \( 1 + 7.16iT - 19T^{2} \)
23 \( 1 + (-4.73 - 4.73i)T + 23iT^{2} \)
29 \( 1 + (4.79 - 4.79i)T - 29iT^{2} \)
31 \( 1 + (-3.40 + 3.40i)T - 31iT^{2} \)
37 \( 1 + (1.37 - 1.37i)T - 37iT^{2} \)
41 \( 1 + (-1.66 - 1.66i)T + 41iT^{2} \)
43 \( 1 - 11.7iT - 43T^{2} \)
47 \( 1 + 1.65T + 47T^{2} \)
53 \( 1 + 6.81iT - 53T^{2} \)
59 \( 1 - 0.484iT - 59T^{2} \)
61 \( 1 + (-4.86 - 4.86i)T + 61iT^{2} \)
67 \( 1 + 1.87T + 67T^{2} \)
71 \( 1 + (-1.21 + 1.21i)T - 71iT^{2} \)
73 \( 1 + (-0.202 + 0.202i)T - 73iT^{2} \)
79 \( 1 + (-3.80 - 3.80i)T + 79iT^{2} \)
83 \( 1 - 9.94iT - 83T^{2} \)
89 \( 1 + 4.30T + 89T^{2} \)
97 \( 1 + (-9.01 + 9.01i)T - 97iT^{2} \)
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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

−14.68612910734167593333995276869, −13.61531947705170210897344216945, −12.90048161249393624089736919365, −11.33766067025112354811655048670, −9.444125492648529809661950699553, −8.266036093035617607070625286636, −7.47767565254877150870191224753, −6.82466121839099380516016418398, −5.19449471782889344578460285821, −2.86124431411402519785983983850, 2.50606871569465001846473175196, 3.72085121446463978584232695108, 4.89236162483093334867968802127, 7.66026382121540309232587253662, 9.000528741146007901423530254936, 9.916671992158567678504254953536, 10.45800332988412824064329003933, 12.03557549874680266361706248384, 12.64170636757837624318781762062, 14.06128522225399863656765103072

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