Properties

Label 2-8034-1.1-c1-0-158
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 + 3.54·5-s + 6-s + 0.812·7-s − 8-s + 9-s − 3.54·10-s − 0.499·11-s − 12-s − 13-s − 0.812·14-s − 3.54·15-s + 16-s − 2.03·17-s − 18-s − 5.19·19-s + 3.54·20-s − 0.812·21-s + 0.499·22-s − 5.38·23-s + 24-s + 7.59·25-s + 26-s − 27-s + 0.812·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.58·5-s + 0.408·6-s + 0.307·7-s − 0.353·8-s + 0.333·9-s − 1.12·10-s − 0.150·11-s − 0.288·12-s − 0.277·13-s − 0.217·14-s − 0.916·15-s + 0.250·16-s − 0.493·17-s − 0.235·18-s − 1.19·19-s + 0.793·20-s − 0.177·21-s + 0.106·22-s − 1.12·23-s + 0.204·24-s + 1.51·25-s + 0.196·26-s − 0.192·27-s + 0.153·28-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 \( 1 - 3.54T + 5T^{2} \)
7 \( 1 - 0.812T + 7T^{2} \)
11 \( 1 + 0.499T + 11T^{2} \)
17 \( 1 + 2.03T + 17T^{2} \)
19 \( 1 + 5.19T + 19T^{2} \)
23 \( 1 + 5.38T + 23T^{2} \)
29 \( 1 - 2.20T + 29T^{2} \)
31 \( 1 + 3.36T + 31T^{2} \)
37 \( 1 + 1.41T + 37T^{2} \)
41 \( 1 - 3.63T + 41T^{2} \)
43 \( 1 - 9.88T + 43T^{2} \)
47 \( 1 + 2.07T + 47T^{2} \)
53 \( 1 - 8.30T + 53T^{2} \)
59 \( 1 + 12.9T + 59T^{2} \)
61 \( 1 - 12.1T + 61T^{2} \)
67 \( 1 + 8.68T + 67T^{2} \)
71 \( 1 + 1.09T + 71T^{2} \)
73 \( 1 + 8.30T + 73T^{2} \)
79 \( 1 - 2.73T + 79T^{2} \)
83 \( 1 - 4.12T + 83T^{2} \)
89 \( 1 + 7.94T + 89T^{2} \)
97 \( 1 + 7.10T + 97T^{2} \)
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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

−7.45207959309855267387539549981, −6.68334202393239039976151640750, −6.10527312013851282161005883145, −5.66979270974193214557113416304, −4.83384842979677336635549111762, −4.02909323684326606156900210997, −2.62785816698126487349260220144, −2.08069407542178086001755486707, −1.32668120595143556679473235976, 0, 1.32668120595143556679473235976, 2.08069407542178086001755486707, 2.62785816698126487349260220144, 4.02909323684326606156900210997, 4.83384842979677336635549111762, 5.66979270974193214557113416304, 6.10527312013851282161005883145, 6.68334202393239039976151640750, 7.45207959309855267387539549981

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