Properties

Label 2-4012-1.1-c1-0-51
Degree $2$
Conductor $4012$
Sign $-1$
Analytic cond. $32.0359$
Root an. cond. $5.66003$
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  − 3.40·3-s + 1.59·5-s + 2.94·7-s + 8.62·9-s − 4.17·11-s + 2.42·13-s − 5.42·15-s + 17-s + 2.92·19-s − 10.0·21-s − 5.49·23-s − 2.46·25-s − 19.1·27-s + 6.57·29-s − 9.00·31-s + 14.2·33-s + 4.68·35-s − 9.65·37-s − 8.27·39-s − 6.88·41-s + 8.59·43-s + 13.7·45-s − 9.92·47-s + 1.67·49-s − 3.40·51-s + 7.14·53-s − 6.63·55-s + ⋯
L(s)  = 1  − 1.96·3-s + 0.711·5-s + 1.11·7-s + 2.87·9-s − 1.25·11-s + 0.673·13-s − 1.40·15-s + 0.242·17-s + 0.670·19-s − 2.19·21-s − 1.14·23-s − 0.493·25-s − 3.68·27-s + 1.22·29-s − 1.61·31-s + 2.47·33-s + 0.792·35-s − 1.58·37-s − 1.32·39-s − 1.07·41-s + 1.31·43-s + 2.04·45-s − 1.44·47-s + 0.239·49-s − 0.477·51-s + 0.982·53-s − 0.895·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4012\)    =    \(2^{2} \cdot 17 \cdot 59\)
Sign: $-1$
Analytic conductor: \(32.0359\)
Root analytic conductor: \(5.66003\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4012,\ (\ :1/2),\ -1)\)

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 \( 1 \)
17 \( 1 - T \)
59 \( 1 + T \)
good3 \( 1 + 3.40T + 3T^{2} \)
5 \( 1 - 1.59T + 5T^{2} \)
7 \( 1 - 2.94T + 7T^{2} \)
11 \( 1 + 4.17T + 11T^{2} \)
13 \( 1 - 2.42T + 13T^{2} \)
19 \( 1 - 2.92T + 19T^{2} \)
23 \( 1 + 5.49T + 23T^{2} \)
29 \( 1 - 6.57T + 29T^{2} \)
31 \( 1 + 9.00T + 31T^{2} \)
37 \( 1 + 9.65T + 37T^{2} \)
41 \( 1 + 6.88T + 41T^{2} \)
43 \( 1 - 8.59T + 43T^{2} \)
47 \( 1 + 9.92T + 47T^{2} \)
53 \( 1 - 7.14T + 53T^{2} \)
61 \( 1 - 2.68T + 61T^{2} \)
67 \( 1 + 12.4T + 67T^{2} \)
71 \( 1 + 2.57T + 71T^{2} \)
73 \( 1 + 14.5T + 73T^{2} \)
79 \( 1 - 5.62T + 79T^{2} \)
83 \( 1 - 11.7T + 83T^{2} \)
89 \( 1 - 3.27T + 89T^{2} \)
97 \( 1 - 5.72T + 97T^{2} \)
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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

−7.80726343786998312902509955926, −7.31917024212943245582576232782, −6.31795561090904505738182216258, −5.74033925271275853975835088745, −5.21504523944744937210038007555, −4.72737649941295110487915843913, −3.66347384709213451686146777643, −2.00945012594455727412844236683, −1.33644365673593722320694075076, 0, 1.33644365673593722320694075076, 2.00945012594455727412844236683, 3.66347384709213451686146777643, 4.72737649941295110487915843913, 5.21504523944744937210038007555, 5.74033925271275853975835088745, 6.31795561090904505738182216258, 7.31917024212943245582576232782, 7.80726343786998312902509955926

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