Properties

Label 2-4024-1.1-c1-0-121
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.11·3-s + 0.544·5-s − 4.06·7-s + 6.70·9-s − 2.60·11-s − 2.55·13-s + 1.69·15-s − 7.12·17-s − 0.832·19-s − 12.6·21-s + 0.370·23-s − 4.70·25-s + 11.5·27-s + 0.116·29-s + 4.33·31-s − 8.12·33-s − 2.21·35-s + 10.0·37-s − 7.95·39-s − 7.49·41-s − 11.6·43-s + 3.64·45-s + 4.33·47-s + 9.53·49-s − 22.1·51-s − 0.218·53-s − 1.41·55-s + ⋯
L(s)  = 1  + 1.79·3-s + 0.243·5-s − 1.53·7-s + 2.23·9-s − 0.786·11-s − 0.708·13-s + 0.437·15-s − 1.72·17-s − 0.190·19-s − 2.76·21-s + 0.0771·23-s − 0.940·25-s + 2.21·27-s + 0.0215·29-s + 0.778·31-s − 1.41·33-s − 0.373·35-s + 1.65·37-s − 1.27·39-s − 1.17·41-s − 1.78·43-s + 0.543·45-s + 0.632·47-s + 1.36·49-s − 3.10·51-s − 0.0300·53-s − 0.191·55-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: \(1\)
Selberg data: \((2,\ 4024,\ (\ :1/2),\ -1)\)

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 \( 1 \)
503 \( 1 + T \)
good3 \( 1 - 3.11T + 3T^{2} \)
5 \( 1 - 0.544T + 5T^{2} \)
7 \( 1 + 4.06T + 7T^{2} \)
11 \( 1 + 2.60T + 11T^{2} \)
13 \( 1 + 2.55T + 13T^{2} \)
17 \( 1 + 7.12T + 17T^{2} \)
19 \( 1 + 0.832T + 19T^{2} \)
23 \( 1 - 0.370T + 23T^{2} \)
29 \( 1 - 0.116T + 29T^{2} \)
31 \( 1 - 4.33T + 31T^{2} \)
37 \( 1 - 10.0T + 37T^{2} \)
41 \( 1 + 7.49T + 41T^{2} \)
43 \( 1 + 11.6T + 43T^{2} \)
47 \( 1 - 4.33T + 47T^{2} \)
53 \( 1 + 0.218T + 53T^{2} \)
59 \( 1 + 8.73T + 59T^{2} \)
61 \( 1 + 2.76T + 61T^{2} \)
67 \( 1 + 6.35T + 67T^{2} \)
71 \( 1 + 12.8T + 71T^{2} \)
73 \( 1 + 0.0547T + 73T^{2} \)
79 \( 1 - 7.48T + 79T^{2} \)
83 \( 1 + 1.01T + 83T^{2} \)
89 \( 1 + 2.49T + 89T^{2} \)
97 \( 1 - 0.239T + 97T^{2} \)
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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

−8.149893056208596613357610808458, −7.46902156884597687254375827516, −6.74151830006191241700618961322, −6.12853221977121903334272623022, −4.78650011310683425809925137057, −4.07656590174164367404145698883, −3.11513035062330318364908076002, −2.65730447183787551906462236810, −1.90372766758960153934335183367, 0, 1.90372766758960153934335183367, 2.65730447183787551906462236810, 3.11513035062330318364908076002, 4.07656590174164367404145698883, 4.78650011310683425809925137057, 6.12853221977121903334272623022, 6.74151830006191241700618961322, 7.46902156884597687254375827516, 8.149893056208596613357610808458

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