Properties

Label 2-8015-1.1-c1-0-301
Degree $2$
Conductor $8015$
Sign $-1$
Analytic cond. $64.0000$
Root an. cond. $8.00000$
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  + 1.00·2-s − 1.61·3-s − 0.983·4-s + 5-s − 1.63·6-s + 7-s − 3.00·8-s − 0.379·9-s + 1.00·10-s − 4.04·11-s + 1.59·12-s − 1.03·13-s + 1.00·14-s − 1.61·15-s − 1.06·16-s − 0.400·17-s − 0.382·18-s − 0.496·19-s − 0.983·20-s − 1.61·21-s − 4.07·22-s + 5.90·23-s + 4.86·24-s + 25-s − 1.03·26-s + 5.47·27-s − 0.983·28-s + ⋯
L(s)  = 1  + 0.713·2-s − 0.934·3-s − 0.491·4-s + 0.447·5-s − 0.666·6-s + 0.377·7-s − 1.06·8-s − 0.126·9-s + 0.318·10-s − 1.21·11-s + 0.459·12-s − 0.285·13-s + 0.269·14-s − 0.417·15-s − 0.266·16-s − 0.0972·17-s − 0.0902·18-s − 0.113·19-s − 0.219·20-s − 0.353·21-s − 0.868·22-s + 1.23·23-s + 0.994·24-s + 0.200·25-s − 0.203·26-s + 1.05·27-s − 0.185·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8015\)    =    \(5 \cdot 7 \cdot 229\)
Sign: $-1$
Analytic conductor: \(64.0000\)
Root analytic conductor: \(8.00000\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8015,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 - T \)
229 \( 1 + T \)
good2 \( 1 - 1.00T + 2T^{2} \)
3 \( 1 + 1.61T + 3T^{2} \)
11 \( 1 + 4.04T + 11T^{2} \)
13 \( 1 + 1.03T + 13T^{2} \)
17 \( 1 + 0.400T + 17T^{2} \)
19 \( 1 + 0.496T + 19T^{2} \)
23 \( 1 - 5.90T + 23T^{2} \)
29 \( 1 + 3.94T + 29T^{2} \)
31 \( 1 - 7.68T + 31T^{2} \)
37 \( 1 - 10.0T + 37T^{2} \)
41 \( 1 - 4.56T + 41T^{2} \)
43 \( 1 - 6.96T + 43T^{2} \)
47 \( 1 + 7.20T + 47T^{2} \)
53 \( 1 + 5.48T + 53T^{2} \)
59 \( 1 + 12.9T + 59T^{2} \)
61 \( 1 - 3.68T + 61T^{2} \)
67 \( 1 + 12.4T + 67T^{2} \)
71 \( 1 - 11.8T + 71T^{2} \)
73 \( 1 + 9.91T + 73T^{2} \)
79 \( 1 + 3.76T + 79T^{2} \)
83 \( 1 - 3.48T + 83T^{2} \)
89 \( 1 - 11.8T + 89T^{2} \)
97 \( 1 + 17.0T + 97T^{2} \)
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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

−7.48260397815280103455879222205, −6.33269144506746260346251861099, −6.03700107128424206677096921089, −5.20372172447162899720328130262, −4.90704213933219523229875529015, −4.25660700021896039731897289384, −2.99961658033716002974446869582, −2.55816467587028229698376962066, −1.05128661818663000051710967037, 0, 1.05128661818663000051710967037, 2.55816467587028229698376962066, 2.99961658033716002974446869582, 4.25660700021896039731897289384, 4.90704213933219523229875529015, 5.20372172447162899720328130262, 6.03700107128424206677096921089, 6.33269144506746260346251861099, 7.48260397815280103455879222205

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