Properties

Label 2-4012-1.1-c1-0-24
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.50·3-s + 2.93·5-s − 0.515·7-s + 3.27·9-s + 4.74·11-s + 0.532·13-s − 7.36·15-s + 17-s + 4.25·19-s + 1.29·21-s + 8.33·23-s + 3.64·25-s − 0.680·27-s − 2.15·29-s + 0.646·31-s − 11.8·33-s − 1.51·35-s − 5.29·37-s − 1.33·39-s − 5.05·41-s + 7.82·43-s + 9.61·45-s + 7.28·47-s − 6.73·49-s − 2.50·51-s + 3.75·53-s + 13.9·55-s + ⋯
L(s)  = 1  − 1.44·3-s + 1.31·5-s − 0.194·7-s + 1.09·9-s + 1.43·11-s + 0.147·13-s − 1.90·15-s + 0.242·17-s + 0.975·19-s + 0.281·21-s + 1.73·23-s + 0.728·25-s − 0.131·27-s − 0.400·29-s + 0.116·31-s − 2.06·33-s − 0.256·35-s − 0.870·37-s − 0.213·39-s − 0.789·41-s + 1.19·43-s + 1.43·45-s + 1.06·47-s − 0.962·49-s − 0.350·51-s + 0.515·53-s + 1.88·55-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: \(0\)
Selberg data: \((2,\ 4012,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.764096547\)
\(L(\frac12)\) \(\approx\) \(1.764096547\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.50T + 3T^{2} \)
5 \( 1 - 2.93T + 5T^{2} \)
7 \( 1 + 0.515T + 7T^{2} \)
11 \( 1 - 4.74T + 11T^{2} \)
13 \( 1 - 0.532T + 13T^{2} \)
19 \( 1 - 4.25T + 19T^{2} \)
23 \( 1 - 8.33T + 23T^{2} \)
29 \( 1 + 2.15T + 29T^{2} \)
31 \( 1 - 0.646T + 31T^{2} \)
37 \( 1 + 5.29T + 37T^{2} \)
41 \( 1 + 5.05T + 41T^{2} \)
43 \( 1 - 7.82T + 43T^{2} \)
47 \( 1 - 7.28T + 47T^{2} \)
53 \( 1 - 3.75T + 53T^{2} \)
61 \( 1 - 8.88T + 61T^{2} \)
67 \( 1 + 4.98T + 67T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 + 0.881T + 73T^{2} \)
79 \( 1 - 10.4T + 79T^{2} \)
83 \( 1 - 1.63T + 83T^{2} \)
89 \( 1 + 8.25T + 89T^{2} \)
97 \( 1 - 5.34T + 97T^{2} \)
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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

−8.740952649713837416744050063923, −7.31479075716033165878259089634, −6.75388634561921720109331035223, −6.17872495274548029048924397455, −5.50256283162977469399114658910, −5.06684497418618647405515212494, −3.99587756676030284544670979076, −2.92992343573165932560870590795, −1.59164591217644595215340025981, −0.908473046826744783819011805578, 0.908473046826744783819011805578, 1.59164591217644595215340025981, 2.92992343573165932560870590795, 3.99587756676030284544670979076, 5.06684497418618647405515212494, 5.50256283162977469399114658910, 6.17872495274548029048924397455, 6.75388634561921720109331035223, 7.31479075716033165878259089634, 8.740952649713837416744050063923

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