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Properties

Label 2-85-5.4-c1-0-1
Degree $2$
Conductor $85$
Sign $0.997 - 0.0654i$
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  − 1.57i·2-s + 2.87i·3-s − 0.494·4-s + (0.146 + 2.23i)5-s + 4.53·6-s − 1.42i·7-s − 2.37i·8-s − 5.24·9-s + (3.52 − 0.231i)10-s + 0.0740·11-s − 1.42i·12-s − 5.70i·13-s − 2.24·14-s + (−6.40 + 0.420i)15-s − 4.74·16-s + i·17-s + ⋯
L(s)  = 1  − 1.11i·2-s + 1.65i·3-s − 0.247·4-s + (0.0654 + 0.997i)5-s + 1.85·6-s − 0.536i·7-s − 0.840i·8-s − 1.74·9-s + (1.11 − 0.0731i)10-s + 0.0223·11-s − 0.410i·12-s − 1.58i·13-s − 0.599·14-s + (−1.65 + 0.108i)15-s − 1.18·16-s + 0.242i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $0.997 - 0.0654i$
Analytic conductor: \(0.678728\)
Root analytic conductor: \(0.823849\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 85,\ (\ :1/2),\ 0.997 - 0.0654i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01553 + 0.0332860i\)
\(L(\frac12)\) \(\approx\) \(1.01553 + 0.0332860i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.146 - 2.23i)T \)
17 \( 1 - iT \)
good2 \( 1 + 1.57iT - 2T^{2} \)
3 \( 1 - 2.87iT - 3T^{2} \)
7 \( 1 + 1.42iT - 7T^{2} \)
11 \( 1 - 0.0740T + 11T^{2} \)
13 \( 1 + 5.70iT - 13T^{2} \)
19 \( 1 - 4.90T + 19T^{2} \)
23 \( 1 - 3.88iT - 23T^{2} \)
29 \( 1 + 5.91T + 29T^{2} \)
31 \( 1 - 0.388T + 31T^{2} \)
37 \( 1 + 9.91iT - 37T^{2} \)
41 \( 1 + 6.61T + 41T^{2} \)
43 \( 1 - 6.94iT - 43T^{2} \)
47 \( 1 - 5.70iT - 47T^{2} \)
53 \( 1 + 0.0216iT - 53T^{2} \)
59 \( 1 - 2T + 59T^{2} \)
61 \( 1 + 3.47T + 61T^{2} \)
67 \( 1 - 6.71iT - 67T^{2} \)
71 \( 1 - 3.84T + 71T^{2} \)
73 \( 1 - 13.5iT - 73T^{2} \)
79 \( 1 - 1.05T + 79T^{2} \)
83 \( 1 + 11.8iT - 83T^{2} \)
89 \( 1 - 1.99T + 89T^{2} \)
97 \( 1 + 5.34iT - 97T^{2} \)
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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

−14.39290144291424038432032453265, −13.10973978871794805504059537895, −11.55171301052548810113526768587, −10.80529398385985662947645173805, −10.19609308384168762839618956373, −9.451939587445893698388504674552, −7.49911545730733127152346504893, −5.60250454769964844469220937588, −3.83429780120465818778217711126, −3.05165080139244125986926659785, 1.93659467977269682117792101212, 5.15909887959126639872194083157, 6.32858783174428522241835341773, 7.21972602277824192909256680897, 8.281939640133313509184731407968, 9.135930178200450976256179791144, 11.67804568240630967046975780479, 12.09264939637504968630227480861, 13.45918334354064281070858125900, 14.01984473898385951141564077387

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