Properties

Label 2-4024-1.1-c1-0-11
Degree $2$
Conductor $4024$
Sign $1$
Analytic cond. $32.1318$
Root an. cond. $5.66849$
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  − 2.23·3-s − 1.90·5-s + 0.0272·7-s + 1.98·9-s − 3.42·11-s + 6.41·13-s + 4.24·15-s + 0.693·17-s − 6.06·19-s − 0.0608·21-s + 2.26·23-s − 1.38·25-s + 2.25·27-s − 0.515·29-s − 9.76·31-s + 7.65·33-s − 0.0517·35-s + 2.77·37-s − 14.3·39-s − 3.95·41-s + 4.04·43-s − 3.78·45-s − 3.28·47-s − 6.99·49-s − 1.54·51-s + 0.768·53-s + 6.51·55-s + ⋯
L(s)  = 1  − 1.28·3-s − 0.849·5-s + 0.0103·7-s + 0.663·9-s − 1.03·11-s + 1.77·13-s + 1.09·15-s + 0.168·17-s − 1.39·19-s − 0.0132·21-s + 0.471·23-s − 0.277·25-s + 0.434·27-s − 0.0957·29-s − 1.75·31-s + 1.33·33-s − 0.00875·35-s + 0.456·37-s − 2.29·39-s − 0.617·41-s + 0.617·43-s − 0.563·45-s − 0.479·47-s − 0.999·49-s − 0.216·51-s + 0.105·53-s + 0.878·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $1$
Analytic conductor: \(32.1318\)
Root analytic conductor: \(5.66849\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4024,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5273378764\)
\(L(\frac12)\) \(\approx\) \(0.5273378764\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 + T \)
good3 \( 1 + 2.23T + 3T^{2} \)
5 \( 1 + 1.90T + 5T^{2} \)
7 \( 1 - 0.0272T + 7T^{2} \)
11 \( 1 + 3.42T + 11T^{2} \)
13 \( 1 - 6.41T + 13T^{2} \)
17 \( 1 - 0.693T + 17T^{2} \)
19 \( 1 + 6.06T + 19T^{2} \)
23 \( 1 - 2.26T + 23T^{2} \)
29 \( 1 + 0.515T + 29T^{2} \)
31 \( 1 + 9.76T + 31T^{2} \)
37 \( 1 - 2.77T + 37T^{2} \)
41 \( 1 + 3.95T + 41T^{2} \)
43 \( 1 - 4.04T + 43T^{2} \)
47 \( 1 + 3.28T + 47T^{2} \)
53 \( 1 - 0.768T + 53T^{2} \)
59 \( 1 - 5.88T + 59T^{2} \)
61 \( 1 + 2.19T + 61T^{2} \)
67 \( 1 + 5.70T + 67T^{2} \)
71 \( 1 + 5.79T + 71T^{2} \)
73 \( 1 + 6.45T + 73T^{2} \)
79 \( 1 + 13.7T + 79T^{2} \)
83 \( 1 - 8.22T + 83T^{2} \)
89 \( 1 - 7.71T + 89T^{2} \)
97 \( 1 - 9.17T + 97T^{2} \)
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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

−8.400905372751344124077112955284, −7.68706631371156299995419716002, −6.87986704768066199669828517226, −6.06859657227910600508866847747, −5.63198104694227227509811610835, −4.72509823750706495053278144992, −3.98235701321387874224261656932, −3.14676078721234361237637176528, −1.72910442062650810068365274019, −0.44154981040534080255886940424, 0.44154981040534080255886940424, 1.72910442062650810068365274019, 3.14676078721234361237637176528, 3.98235701321387874224261656932, 4.72509823750706495053278144992, 5.63198104694227227509811610835, 6.06859657227910600508866847747, 6.87986704768066199669828517226, 7.68706631371156299995419716002, 8.400905372751344124077112955284

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