Properties

Label 2-8034-1.1-c1-0-41
Degree $2$
Conductor $8034$
Sign $-1$
Analytic cond. $64.1518$
Root an. cond. $8.00948$
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-s − 3-s + 4-s − 2.24·5-s + 6-s − 4.80·7-s − 8-s + 9-s + 2.24·10-s − 5.91·11-s − 12-s − 13-s + 4.80·14-s + 2.24·15-s + 16-s + 0.672·17-s − 18-s − 6.07·19-s − 2.24·20-s + 4.80·21-s + 5.91·22-s − 1.49·23-s + 24-s + 0.0270·25-s + 26-s − 27-s − 4.80·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.00·5-s + 0.408·6-s − 1.81·7-s − 0.353·8-s + 0.333·9-s + 0.709·10-s − 1.78·11-s − 0.288·12-s − 0.277·13-s + 1.28·14-s + 0.578·15-s + 0.250·16-s + 0.163·17-s − 0.235·18-s − 1.39·19-s − 0.501·20-s + 1.04·21-s + 1.26·22-s − 0.311·23-s + 0.204·24-s + 0.00540·25-s + 0.196·26-s − 0.192·27-s − 0.908·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8034\)    =    \(2 \cdot 3 \cdot 13 \cdot 103\)
Sign: $-1$
Analytic conductor: \(64.1518\)
Root analytic conductor: \(8.00948\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8034,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 + T \)
3 \( 1 + T \)
13 \( 1 + T \)
103 \( 1 + T \)
good5 \( 1 + 2.24T + 5T^{2} \)
7 \( 1 + 4.80T + 7T^{2} \)
11 \( 1 + 5.91T + 11T^{2} \)
17 \( 1 - 0.672T + 17T^{2} \)
19 \( 1 + 6.07T + 19T^{2} \)
23 \( 1 + 1.49T + 23T^{2} \)
29 \( 1 - 4.90T + 29T^{2} \)
31 \( 1 + 0.627T + 31T^{2} \)
37 \( 1 - 9.51T + 37T^{2} \)
41 \( 1 - 4.06T + 41T^{2} \)
43 \( 1 - 8.49T + 43T^{2} \)
47 \( 1 - 3.45T + 47T^{2} \)
53 \( 1 - 7.99T + 53T^{2} \)
59 \( 1 - 4.41T + 59T^{2} \)
61 \( 1 + 5.58T + 61T^{2} \)
67 \( 1 + 0.920T + 67T^{2} \)
71 \( 1 + 1.63T + 71T^{2} \)
73 \( 1 + 13.4T + 73T^{2} \)
79 \( 1 + 6.26T + 79T^{2} \)
83 \( 1 + 5.91T + 83T^{2} \)
89 \( 1 - 8.71T + 89T^{2} \)
97 \( 1 + 4.09T + 97T^{2} \)
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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

−7.45104492673049295233442415851, −6.99178823034219815343357071351, −6.04578975265321020891992859833, −5.78177605654444875082346768264, −4.53901021289214809994491845581, −3.92362781766504775664671779288, −2.86440448330526841417721886344, −2.44542128339835244499891040466, −0.64238703580943644630865332956, 0, 0.64238703580943644630865332956, 2.44542128339835244499891040466, 2.86440448330526841417721886344, 3.92362781766504775664671779288, 4.53901021289214809994491845581, 5.78177605654444875082346768264, 6.04578975265321020891992859833, 6.99178823034219815343357071351, 7.45104492673049295233442415851

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