Properties

Label 2-4034-1.1-c1-0-28
Degree $2$
Conductor $4034$
Sign $1$
Analytic cond. $32.2116$
Root an. cond. $5.67553$
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 − 0.128·3-s + 4-s + 3.87·5-s + 0.128·6-s − 4.13·7-s − 8-s − 2.98·9-s − 3.87·10-s + 0.767·11-s − 0.128·12-s − 4.27·13-s + 4.13·14-s − 0.496·15-s + 16-s − 3.18·17-s + 2.98·18-s + 5.33·19-s + 3.87·20-s + 0.530·21-s − 0.767·22-s − 0.226·23-s + 0.128·24-s + 10.0·25-s + 4.27·26-s + 0.766·27-s − 4.13·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.0739·3-s + 0.5·4-s + 1.73·5-s + 0.0523·6-s − 1.56·7-s − 0.353·8-s − 0.994·9-s − 1.22·10-s + 0.231·11-s − 0.0369·12-s − 1.18·13-s + 1.10·14-s − 0.128·15-s + 0.250·16-s − 0.772·17-s + 0.703·18-s + 1.22·19-s + 0.866·20-s + 0.115·21-s − 0.163·22-s − 0.0471·23-s + 0.0261·24-s + 2.00·25-s + 0.838·26-s + 0.147·27-s − 0.781·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4034\)    =    \(2 \cdot 2017\)
Sign: $1$
Analytic conductor: \(32.2116\)
Root analytic conductor: \(5.67553\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4034,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.143645868\)
\(L(\frac12)\) \(\approx\) \(1.143645868\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
2017 \( 1 - T \)
good3 \( 1 + 0.128T + 3T^{2} \)
5 \( 1 - 3.87T + 5T^{2} \)
7 \( 1 + 4.13T + 7T^{2} \)
11 \( 1 - 0.767T + 11T^{2} \)
13 \( 1 + 4.27T + 13T^{2} \)
17 \( 1 + 3.18T + 17T^{2} \)
19 \( 1 - 5.33T + 19T^{2} \)
23 \( 1 + 0.226T + 23T^{2} \)
29 \( 1 - 0.979T + 29T^{2} \)
31 \( 1 + 1.08T + 31T^{2} \)
37 \( 1 + 2.26T + 37T^{2} \)
41 \( 1 - 3.62T + 41T^{2} \)
43 \( 1 + 7.91T + 43T^{2} \)
47 \( 1 - 7.45T + 47T^{2} \)
53 \( 1 - 9.72T + 53T^{2} \)
59 \( 1 + 3.40T + 59T^{2} \)
61 \( 1 + 11.5T + 61T^{2} \)
67 \( 1 - 9.10T + 67T^{2} \)
71 \( 1 - 12.7T + 71T^{2} \)
73 \( 1 - 0.693T + 73T^{2} \)
79 \( 1 - 15.2T + 79T^{2} \)
83 \( 1 - 15.1T + 83T^{2} \)
89 \( 1 - 11.0T + 89T^{2} \)
97 \( 1 - 1.00T + 97T^{2} \)
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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

−8.811411515084584532634216495684, −7.67425534378059733971612283481, −6.73467935905905562000978883641, −6.42205766090067310423959778133, −5.62524209536951199905871252542, −5.06187066576645622409514612227, −3.45761948857249659541410677965, −2.67143606473108740340242420416, −2.10771113075467464670789459463, −0.64818029210386076059259531321, 0.64818029210386076059259531321, 2.10771113075467464670789459463, 2.67143606473108740340242420416, 3.45761948857249659541410677965, 5.06187066576645622409514612227, 5.62524209536951199905871252542, 6.42205766090067310423959778133, 6.73467935905905562000978883641, 7.67425534378059733971612283481, 8.811411515084584532634216495684

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