Properties

Degree 2
Conductor $ 2^{2} \cdot 17 \cdot 59 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.692·3-s − 3.58·5-s + 3.81·7-s − 2.52·9-s − 3.20·11-s + 3.92·13-s − 2.48·15-s − 17-s + 0.300·19-s + 2.63·21-s − 1.11·23-s + 7.84·25-s − 3.82·27-s + 7.45·29-s + 0.368·31-s − 2.21·33-s − 13.6·35-s − 5.36·37-s + 2.71·39-s + 0.470·41-s + 8.69·43-s + 9.03·45-s − 8.28·47-s + 7.52·49-s − 0.692·51-s − 14.2·53-s + 11.4·55-s + ⋯
L(s)  = 1  + 0.399·3-s − 1.60·5-s + 1.44·7-s − 0.840·9-s − 0.965·11-s + 1.08·13-s − 0.641·15-s − 0.242·17-s + 0.0689·19-s + 0.576·21-s − 0.233·23-s + 1.56·25-s − 0.735·27-s + 1.38·29-s + 0.0661·31-s − 0.386·33-s − 2.30·35-s − 0.882·37-s + 0.435·39-s + 0.0734·41-s + 1.32·43-s + 1.34·45-s − 1.20·47-s + 1.07·49-s − 0.0970·51-s − 1.96·53-s + 1.54·55-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

\( d \)  =  \(2\)
\( N \)  =  \(4012\)    =    \(2^{2} \cdot 17 \cdot 59\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4012} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 4012,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;17,\;59\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;17,\;59\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
17 \( 1 + T \)
59 \( 1 - T \)
good3 \( 1 - 0.692T + 3T^{2} \)
5 \( 1 + 3.58T + 5T^{2} \)
7 \( 1 - 3.81T + 7T^{2} \)
11 \( 1 + 3.20T + 11T^{2} \)
13 \( 1 - 3.92T + 13T^{2} \)
19 \( 1 - 0.300T + 19T^{2} \)
23 \( 1 + 1.11T + 23T^{2} \)
29 \( 1 - 7.45T + 29T^{2} \)
31 \( 1 - 0.368T + 31T^{2} \)
37 \( 1 + 5.36T + 37T^{2} \)
41 \( 1 - 0.470T + 41T^{2} \)
43 \( 1 - 8.69T + 43T^{2} \)
47 \( 1 + 8.28T + 47T^{2} \)
53 \( 1 + 14.2T + 53T^{2} \)
61 \( 1 + 2.58T + 61T^{2} \)
67 \( 1 + 6.13T + 67T^{2} \)
71 \( 1 + 5.69T + 71T^{2} \)
73 \( 1 - 0.352T + 73T^{2} \)
79 \( 1 - 0.179T + 79T^{2} \)
83 \( 1 - 11.2T + 83T^{2} \)
89 \( 1 + 16.2T + 89T^{2} \)
97 \( 1 + 13.6T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.109011639820419121419949443774, −7.77909421809421786803340181741, −6.78466811617058959619135918805, −5.74239845866925256254357040559, −4.88552819439562207196391002476, −4.29583037419403156257889047008, −3.40342892521674644137820295354, −2.65424623835679030794815915834, −1.37343934252468851156360781326, 0, 1.37343934252468851156360781326, 2.65424623835679030794815915834, 3.40342892521674644137820295354, 4.29583037419403156257889047008, 4.88552819439562207196391002476, 5.74239845866925256254357040559, 6.78466811617058959619135918805, 7.77909421809421786803340181741, 8.109011639820419121419949443774

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