Properties

Label 2-8030-1.1-c1-0-58
Degree $2$
Conductor $8030$
Sign $1$
Analytic cond. $64.1198$
Root an. cond. $8.00748$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 0.765·3-s + 4-s − 5-s − 0.765·6-s + 2.95·7-s + 8-s − 2.41·9-s − 10-s − 11-s − 0.765·12-s − 2.75·13-s + 2.95·14-s + 0.765·15-s + 16-s + 0.534·17-s − 2.41·18-s + 6.63·19-s − 20-s − 2.26·21-s − 22-s − 6.23·23-s − 0.765·24-s + 25-s − 2.75·26-s + 4.14·27-s + 2.95·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.441·3-s + 0.5·4-s − 0.447·5-s − 0.312·6-s + 1.11·7-s + 0.353·8-s − 0.804·9-s − 0.316·10-s − 0.301·11-s − 0.220·12-s − 0.763·13-s + 0.790·14-s + 0.197·15-s + 0.250·16-s + 0.129·17-s − 0.569·18-s + 1.52·19-s − 0.223·20-s − 0.493·21-s − 0.213·22-s − 1.30·23-s − 0.156·24-s + 0.200·25-s − 0.540·26-s + 0.797·27-s + 0.559·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8030\)    =    \(2 \cdot 5 \cdot 11 \cdot 73\)
Sign: $1$
Analytic conductor: \(64.1198\)
Root analytic conductor: \(8.00748\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8030,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.430278607\)
\(L(\frac12)\) \(\approx\) \(2.430278607\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 + T \)
11 \( 1 + T \)
73 \( 1 + T \)
good3 \( 1 + 0.765T + 3T^{2} \)
7 \( 1 - 2.95T + 7T^{2} \)
13 \( 1 + 2.75T + 13T^{2} \)
17 \( 1 - 0.534T + 17T^{2} \)
19 \( 1 - 6.63T + 19T^{2} \)
23 \( 1 + 6.23T + 23T^{2} \)
29 \( 1 + 5.58T + 29T^{2} \)
31 \( 1 + 0.857T + 31T^{2} \)
37 \( 1 - 7.28T + 37T^{2} \)
41 \( 1 - 7.08T + 41T^{2} \)
43 \( 1 - 8.08T + 43T^{2} \)
47 \( 1 + 2.78T + 47T^{2} \)
53 \( 1 + 5.77T + 53T^{2} \)
59 \( 1 + 5.27T + 59T^{2} \)
61 \( 1 - 2.10T + 61T^{2} \)
67 \( 1 + 3.63T + 67T^{2} \)
71 \( 1 - 11.9T + 71T^{2} \)
79 \( 1 - 9.85T + 79T^{2} \)
83 \( 1 + 1.35T + 83T^{2} \)
89 \( 1 - 6.85T + 89T^{2} \)
97 \( 1 - 13.1T + 97T^{2} \)
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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

−7.77144505337483483391588687386, −7.28981020666279027418083686641, −6.16178354823191983587412843421, −5.65684590479583039018442499613, −5.02319916283556903661134492819, −4.47840234180796269690843245894, −3.59400445811978452828712357197, −2.74200888261616494859248417868, −1.92636314371433718747645005413, −0.69888702259428413170448410024, 0.69888702259428413170448410024, 1.92636314371433718747645005413, 2.74200888261616494859248417868, 3.59400445811978452828712357197, 4.47840234180796269690843245894, 5.02319916283556903661134492819, 5.65684590479583039018442499613, 6.16178354823191983587412843421, 7.28981020666279027418083686641, 7.77144505337483483391588687386

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