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Properties

Label 2-85-17.13-c1-0-0
Degree $2$
Conductor $85$
Sign $-0.554 - 0.831i$
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  + 2.24i·2-s + (0.140 + 0.140i)3-s − 3.05·4-s + (0.707 + 0.707i)5-s + (−0.314 + 0.314i)6-s + (−1.33 + 1.33i)7-s − 2.37i·8-s − 2.96i·9-s + (−1.59 + 1.59i)10-s + (2.49 − 2.49i)11-s + (−0.428 − 0.428i)12-s + 1.27·13-s + (−3.00 − 3.00i)14-s + 0.198i·15-s − 0.766·16-s + (3.68 − 1.85i)17-s + ⋯
L(s)  = 1  + 1.59i·2-s + (0.0808 + 0.0808i)3-s − 1.52·4-s + (0.316 + 0.316i)5-s + (−0.128 + 0.128i)6-s + (−0.505 + 0.505i)7-s − 0.840i·8-s − 0.986i·9-s + (−0.502 + 0.502i)10-s + (0.752 − 0.752i)11-s + (−0.123 − 0.123i)12-s + 0.353·13-s + (−0.804 − 0.804i)14-s + 0.0511i·15-s − 0.191·16-s + (0.893 − 0.449i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 - 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $-0.554 - 0.831i$
Analytic conductor: \(0.678728\)
Root analytic conductor: \(0.823849\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 85,\ (\ :1/2),\ -0.554 - 0.831i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.455219 + 0.850894i\)
\(L(\frac12)\) \(\approx\) \(0.455219 + 0.850894i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.68 + 1.85i)T \)
good2 \( 1 - 2.24iT - 2T^{2} \)
3 \( 1 + (-0.140 - 0.140i)T + 3iT^{2} \)
7 \( 1 + (1.33 - 1.33i)T - 7iT^{2} \)
11 \( 1 + (-2.49 + 2.49i)T - 11iT^{2} \)
13 \( 1 - 1.27T + 13T^{2} \)
19 \( 1 - 4.69iT - 19T^{2} \)
23 \( 1 + (0.406 - 0.406i)T - 23iT^{2} \)
29 \( 1 + (3.81 + 3.81i)T + 29iT^{2} \)
31 \( 1 + (4.39 + 4.39i)T + 31iT^{2} \)
37 \( 1 + (6.00 + 6.00i)T + 37iT^{2} \)
41 \( 1 + (-4.28 + 4.28i)T - 41iT^{2} \)
43 \( 1 - 9.16iT - 43T^{2} \)
47 \( 1 + 10.7T + 47T^{2} \)
53 \( 1 - 9.90iT - 53T^{2} \)
59 \( 1 + 3.15iT - 59T^{2} \)
61 \( 1 + (-3.63 + 3.63i)T - 61iT^{2} \)
67 \( 1 - 0.281T + 67T^{2} \)
71 \( 1 + (-8.30 - 8.30i)T + 71iT^{2} \)
73 \( 1 + (6.95 + 6.95i)T + 73iT^{2} \)
79 \( 1 + (11.9 - 11.9i)T - 79iT^{2} \)
83 \( 1 + 8.51iT - 83T^{2} \)
89 \( 1 - 16.7T + 89T^{2} \)
97 \( 1 + (-4.18 - 4.18i)T + 97iT^{2} \)
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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

−14.59886818189503971948449032541, −14.16117029953212118205610137146, −12.78935191367448227725970528052, −11.49293214487380896386005106896, −9.667232198446189310379683317373, −8.923037312722369323549587128884, −7.61688778401767606367818055690, −6.27695410600013780578984949399, −5.76746245040159108427279443433, −3.65478180832305816676565542493, 1.72159546765021364890788823991, 3.49846203655660322102939909596, 4.93930993355297741718235138730, 6.98930709301607896283576420082, 8.728840947363590486272844720893, 9.834548425445656584725412255905, 10.60198226342928933452582881444, 11.70171166836390207083393930393, 12.81336800667848182933701896683, 13.38285570266283241870087568545

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