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Properties

Label 2-85-17.8-c1-0-4
Degree $2$
Conductor $85$
Sign $0.575 + 0.817i$
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  + (1.86 − 1.86i)2-s + (−0.811 + 1.95i)3-s − 4.96i·4-s + (0.923 + 0.382i)5-s + (2.14 + 5.17i)6-s + (−3.75 + 1.55i)7-s + (−5.54 − 5.54i)8-s + (−1.05 − 1.05i)9-s + (2.43 − 1.01i)10-s + (−0.669 − 1.61i)11-s + (9.73 + 4.03i)12-s + 1.67i·13-s + (−4.10 + 9.90i)14-s + (−1.49 + 1.49i)15-s − 10.7·16-s + (4.08 − 0.593i)17-s + ⋯
L(s)  = 1  + (1.31 − 1.31i)2-s + (−0.468 + 1.13i)3-s − 2.48i·4-s + (0.413 + 0.171i)5-s + (0.874 + 2.11i)6-s + (−1.41 + 0.587i)7-s + (−1.95 − 1.95i)8-s + (−0.352 − 0.352i)9-s + (0.771 − 0.319i)10-s + (−0.201 − 0.487i)11-s + (2.81 + 1.16i)12-s + 0.465i·13-s + (−1.09 + 2.64i)14-s + (−0.387 + 0.387i)15-s − 2.68·16-s + (0.989 − 0.143i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.575 + 0.817i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.575 + 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $0.575 + 0.817i$
Analytic conductor: \(0.678728\)
Root analytic conductor: \(0.823849\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{85} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 85,\ (\ :1/2),\ 0.575 + 0.817i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30547 - 0.677754i\)
\(L(\frac12)\) \(\approx\) \(1.30547 - 0.677754i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.923 - 0.382i)T \)
17 \( 1 + (-4.08 + 0.593i)T \)
good2 \( 1 + (-1.86 + 1.86i)T - 2iT^{2} \)
3 \( 1 + (0.811 - 1.95i)T + (-2.12 - 2.12i)T^{2} \)
7 \( 1 + (3.75 - 1.55i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (0.669 + 1.61i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 - 1.67iT - 13T^{2} \)
19 \( 1 + (0.176 - 0.176i)T - 19iT^{2} \)
23 \( 1 + (0.198 + 0.480i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (-0.449 - 0.186i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (-3.64 + 8.80i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (-1.63 + 3.94i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (4.88 - 2.02i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (2.24 + 2.24i)T + 43iT^{2} \)
47 \( 1 - 6.26iT - 47T^{2} \)
53 \( 1 + (7.24 - 7.24i)T - 53iT^{2} \)
59 \( 1 + (-8.20 - 8.20i)T + 59iT^{2} \)
61 \( 1 + (4.02 - 1.66i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 - 9.73T + 67T^{2} \)
71 \( 1 + (0.384 - 0.929i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-2.69 - 1.11i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (-2.60 - 6.29i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (2.26 - 2.26i)T - 83iT^{2} \)
89 \( 1 + 3.30iT - 89T^{2} \)
97 \( 1 + (-1.57 - 0.650i)T + (68.5 + 68.5i)T^{2} \)
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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

−13.82844186004763060603739462674, −12.92870555134274649998825592335, −11.94437917422245273344557456198, −10.94291433590550052966944057403, −9.950323358450454777000654091748, −9.476752320824542260668823226444, −6.20540050664726219627432346267, −5.41308963773219636302767835462, −4.00316812719343180926738398792, −2.82759175112767052115383124372, 3.38528560722189804279852706497, 5.22439290105399899943628262017, 6.41237696091185860474029726821, 6.92797600139996420168562968742, 8.068040726098050139355007494109, 9.965637173715255001897607953697, 12.05799801705785189147297967939, 12.75417935720215717507962179745, 13.27430405114146876294440199986, 14.15590243962449720430018048700

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