Properties

Label 2-85-1.1-c1-0-2
Degree $2$
Conductor $85$
Sign $-1$
Analytic cond. $0.678728$
Root an. cond. $0.823849$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.41·2-s − 0.585·3-s + 3.82·4-s − 5-s + 1.41·6-s − 3.41·7-s − 4.41·8-s − 2.65·9-s + 2.41·10-s − 5.41·11-s − 2.24·12-s + 2.82·13-s + 8.24·14-s + 0.585·15-s + 2.99·16-s − 17-s + 6.41·18-s + 2.82·19-s − 3.82·20-s + 2·21-s + 13.0·22-s − 0.585·23-s + 2.58·24-s + 25-s − 6.82·26-s + 3.31·27-s − 13.0·28-s + ⋯
L(s)  = 1  − 1.70·2-s − 0.338·3-s + 1.91·4-s − 0.447·5-s + 0.577·6-s − 1.29·7-s − 1.56·8-s − 0.885·9-s + 0.763·10-s − 1.63·11-s − 0.647·12-s + 0.784·13-s + 2.20·14-s + 0.151·15-s + 0.749·16-s − 0.242·17-s + 1.51·18-s + 0.648·19-s − 0.856·20-s + 0.436·21-s + 2.78·22-s − 0.122·23-s + 0.527·24-s + 0.200·25-s − 1.33·26-s + 0.637·27-s − 2.47·28-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + T \)
17 \( 1 + T \)
good2 \( 1 + 2.41T + 2T^{2} \)
3 \( 1 + 0.585T + 3T^{2} \)
7 \( 1 + 3.41T + 7T^{2} \)
11 \( 1 + 5.41T + 11T^{2} \)
13 \( 1 - 2.82T + 13T^{2} \)
19 \( 1 - 2.82T + 19T^{2} \)
23 \( 1 + 0.585T + 23T^{2} \)
29 \( 1 - 0.828T + 29T^{2} \)
31 \( 1 + 4.24T + 31T^{2} \)
37 \( 1 + 10.4T + 37T^{2} \)
41 \( 1 - 10.4T + 41T^{2} \)
43 \( 1 + 3.65T + 43T^{2} \)
47 \( 1 - 0.828T + 47T^{2} \)
53 \( 1 - 11.6T + 53T^{2} \)
59 \( 1 + 14.8T + 59T^{2} \)
61 \( 1 + 3.65T + 61T^{2} \)
67 \( 1 + 8.82T + 67T^{2} \)
71 \( 1 - 4.24T + 71T^{2} \)
73 \( 1 - 0.828T + 73T^{2} \)
79 \( 1 - 2.58T + 79T^{2} \)
83 \( 1 + 13.3T + 83T^{2} \)
89 \( 1 + 13.6T + 89T^{2} \)
97 \( 1 + 7.65T + 97T^{2} \)
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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

−13.58208057480413899838428201851, −12.31775539685803957797677093771, −11.06308369725245175872328519507, −10.39126818503970461796256783247, −9.203714202944556418782199011166, −8.223352963482811590314374917504, −7.10272384563967549403206342880, −5.77708222231259154740652373264, −2.95634545112088224055144157853, 0, 2.95634545112088224055144157853, 5.77708222231259154740652373264, 7.10272384563967549403206342880, 8.223352963482811590314374917504, 9.203714202944556418782199011166, 10.39126818503970461796256783247, 11.06308369725245175872328519507, 12.31775539685803957797677093771, 13.58208057480413899838428201851

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