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Properties

Label 2-85-85.9-c1-0-4
Degree $2$
Conductor $85$
Sign $0.811 - 0.584i$
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  + (1.48 + 1.48i)2-s + (0.690 − 1.66i)3-s + 2.43i·4-s + (−2.22 − 0.184i)5-s + (3.51 − 1.45i)6-s + (−2.47 + 1.02i)7-s + (−0.641 + 0.641i)8-s + (−0.183 − 0.183i)9-s + (−3.04 − 3.59i)10-s + (−0.901 + 0.373i)11-s + (4.05 + 1.67i)12-s − 2.65·13-s + (−5.21 − 2.16i)14-s + (−1.84 + 3.58i)15-s + 2.95·16-s + (3.25 − 2.53i)17-s + ⋯
L(s)  = 1  + (1.05 + 1.05i)2-s + (0.398 − 0.963i)3-s + 1.21i·4-s + (−0.996 − 0.0826i)5-s + (1.43 − 0.593i)6-s + (−0.937 + 0.388i)7-s + (−0.226 + 0.226i)8-s + (−0.0613 − 0.0613i)9-s + (−0.961 − 1.13i)10-s + (−0.271 + 0.112i)11-s + (1.17 + 0.484i)12-s − 0.735·13-s + (−1.39 − 0.577i)14-s + (−0.477 + 0.926i)15-s + 0.738·16-s + (0.789 − 0.613i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 - 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $0.811 - 0.584i$
Analytic conductor: \(0.678728\)
Root analytic conductor: \(0.823849\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 85,\ (\ :1/2),\ 0.811 - 0.584i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38957 + 0.448544i\)
\(L(\frac12)\) \(\approx\) \(1.38957 + 0.448544i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.22 + 0.184i)T \)
17 \( 1 + (-3.25 + 2.53i)T \)
good2 \( 1 + (-1.48 - 1.48i)T + 2iT^{2} \)
3 \( 1 + (-0.690 + 1.66i)T + (-2.12 - 2.12i)T^{2} \)
7 \( 1 + (2.47 - 1.02i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (0.901 - 0.373i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + 2.65T + 13T^{2} \)
19 \( 1 + (0.478 - 0.478i)T - 19iT^{2} \)
23 \( 1 + (-1.54 - 3.72i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (-3.83 + 9.26i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (9.10 + 3.77i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (2.61 - 6.31i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-2.80 - 6.76i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (6.29 - 6.29i)T - 43iT^{2} \)
47 \( 1 - 0.874T + 47T^{2} \)
53 \( 1 + (2.00 + 2.00i)T + 53iT^{2} \)
59 \( 1 + (7.59 + 7.59i)T + 59iT^{2} \)
61 \( 1 + (-0.793 - 1.91i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + 5.73iT - 67T^{2} \)
71 \( 1 + (-1.03 - 0.430i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (-2.77 - 1.14i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (0.377 - 0.156i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (4.64 + 4.64i)T + 83iT^{2} \)
89 \( 1 + 5.62iT - 89T^{2} \)
97 \( 1 + (-14.2 - 5.89i)T + (68.5 + 68.5i)T^{2} \)
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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

−14.37876706353740254326724071275, −13.26333717643990101547181542283, −12.69082900241611087180198253758, −11.78181956390120552672479861346, −9.762949362875318531403801530552, −7.972894800474475780992747159097, −7.39930671162935903341873622925, −6.32171534576200659922573144782, −4.82063475491863613291608046825, −3.18027730371528488875196874763, 3.17106105126081210182067186649, 3.85651403176183235577055423608, 5.07691285919994236489658217944, 7.15904138006511970778056030842, 8.864911233555864448888622015249, 10.30321366215386142747813516720, 10.75770521565847556304430441887, 12.35324468261987737993712564810, 12.67405243118175167539523036655, 14.24757545612069675488364941527

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