Properties

Label 2-4034-1.1-c1-0-48
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 + 2.22·3-s + 4-s − 2.04·5-s − 2.22·6-s + 1.48·7-s − 8-s + 1.93·9-s + 2.04·10-s + 4.55·11-s + 2.22·12-s − 4.47·13-s − 1.48·14-s − 4.54·15-s + 16-s − 4.46·17-s − 1.93·18-s + 6.65·19-s − 2.04·20-s + 3.30·21-s − 4.55·22-s − 2.78·23-s − 2.22·24-s − 0.809·25-s + 4.47·26-s − 2.36·27-s + 1.48·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.28·3-s + 0.5·4-s − 0.915·5-s − 0.907·6-s + 0.561·7-s − 0.353·8-s + 0.645·9-s + 0.647·10-s + 1.37·11-s + 0.641·12-s − 1.23·13-s − 0.397·14-s − 1.17·15-s + 0.250·16-s − 1.08·17-s − 0.456·18-s + 1.52·19-s − 0.457·20-s + 0.720·21-s − 0.972·22-s − 0.580·23-s − 0.453·24-s − 0.161·25-s + 0.876·26-s − 0.454·27-s + 0.280·28-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

\(L(1)\) \(\approx\) \(1.949722174\)
\(L(\frac12)\) \(\approx\) \(1.949722174\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.22T + 3T^{2} \)
5 \( 1 + 2.04T + 5T^{2} \)
7 \( 1 - 1.48T + 7T^{2} \)
11 \( 1 - 4.55T + 11T^{2} \)
13 \( 1 + 4.47T + 13T^{2} \)
17 \( 1 + 4.46T + 17T^{2} \)
19 \( 1 - 6.65T + 19T^{2} \)
23 \( 1 + 2.78T + 23T^{2} \)
29 \( 1 + 1.98T + 29T^{2} \)
31 \( 1 - 5.60T + 31T^{2} \)
37 \( 1 - 11.3T + 37T^{2} \)
41 \( 1 + 6.43T + 41T^{2} \)
43 \( 1 - 5.49T + 43T^{2} \)
47 \( 1 - 13.6T + 47T^{2} \)
53 \( 1 - 11.0T + 53T^{2} \)
59 \( 1 - 10.0T + 59T^{2} \)
61 \( 1 - 5.64T + 61T^{2} \)
67 \( 1 + 0.320T + 67T^{2} \)
71 \( 1 + 6.67T + 71T^{2} \)
73 \( 1 - 7.34T + 73T^{2} \)
79 \( 1 - 17.2T + 79T^{2} \)
83 \( 1 - 3.97T + 83T^{2} \)
89 \( 1 + 8.65T + 89T^{2} \)
97 \( 1 - 10.1T + 97T^{2} \)
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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

−8.331205284729505290999169791943, −7.87666086670009467657042222917, −7.32439434756437226989054460990, −6.64970575818440904484372590185, −5.46037852042880141799485820789, −4.27811097776112440430918893789, −3.84318783262141265640313223481, −2.75134387415706657432158689353, −2.08717736959978380495230312403, −0.839999849746655035498690978880, 0.839999849746655035498690978880, 2.08717736959978380495230312403, 2.75134387415706657432158689353, 3.84318783262141265640313223481, 4.27811097776112440430918893789, 5.46037852042880141799485820789, 6.64970575818440904484372590185, 7.32439434756437226989054460990, 7.87666086670009467657042222917, 8.331205284729505290999169791943

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