Properties

Label 2-4012-1.1-c1-0-30
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  − 3.23·3-s + 0.381·5-s − 3·7-s + 7.47·9-s + 1.61·11-s − 3.61·13-s − 1.23·15-s − 17-s + 2.23·19-s + 9.70·21-s + 2.38·23-s − 4.85·25-s − 14.4·27-s + 5.85·29-s − 9·31-s − 5.23·33-s − 1.14·35-s + 3·37-s + 11.7·39-s + 9·41-s − 1.38·43-s + 2.85·45-s + 3.85·47-s + 2·49-s + 3.23·51-s + 4.70·53-s + 0.618·55-s + ⋯
L(s)  = 1  − 1.86·3-s + 0.170·5-s − 1.13·7-s + 2.49·9-s + 0.487·11-s − 1.00·13-s − 0.319·15-s − 0.242·17-s + 0.512·19-s + 2.11·21-s + 0.496·23-s − 0.970·25-s − 2.78·27-s + 1.08·29-s − 1.61·31-s − 0.911·33-s − 0.193·35-s + 0.493·37-s + 1.87·39-s + 1.40·41-s − 0.210·43-s + 0.425·45-s + 0.562·47-s + 0.285·49-s + 0.453·51-s + 0.646·53-s + 0.0833·55-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 \( 1 + 3.23T + 3T^{2} \)
5 \( 1 - 0.381T + 5T^{2} \)
7 \( 1 + 3T + 7T^{2} \)
11 \( 1 - 1.61T + 11T^{2} \)
13 \( 1 + 3.61T + 13T^{2} \)
19 \( 1 - 2.23T + 19T^{2} \)
23 \( 1 - 2.38T + 23T^{2} \)
29 \( 1 - 5.85T + 29T^{2} \)
31 \( 1 + 9T + 31T^{2} \)
37 \( 1 - 3T + 37T^{2} \)
41 \( 1 - 9T + 41T^{2} \)
43 \( 1 + 1.38T + 43T^{2} \)
47 \( 1 - 3.85T + 47T^{2} \)
53 \( 1 - 4.70T + 53T^{2} \)
61 \( 1 - 7.47T + 61T^{2} \)
67 \( 1 + 10.7T + 67T^{2} \)
71 \( 1 - 13.0T + 71T^{2} \)
73 \( 1 - 9.18T + 73T^{2} \)
79 \( 1 - 0.0901T + 79T^{2} \)
83 \( 1 + 5.47T + 83T^{2} \)
89 \( 1 - 12.3T + 89T^{2} \)
97 \( 1 + 4.90T + 97T^{2} \)
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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

−7.74880174501418480416351622482, −7.02141969021477093672704730020, −6.57216044732979714779857611362, −5.82857253152736941345570098221, −5.30079120533470527014894192282, −4.44207119154618631833142446474, −3.64797509487096491072916419892, −2.34762787278381892146636415303, −1.01168280477197151417615062610, 0, 1.01168280477197151417615062610, 2.34762787278381892146636415303, 3.64797509487096491072916419892, 4.44207119154618631833142446474, 5.30079120533470527014894192282, 5.82857253152736941345570098221, 6.57216044732979714779857611362, 7.02141969021477093672704730020, 7.74880174501418480416351622482

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