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  + 0.660·3-s + 2.45·5-s + 3.10·7-s − 2.56·9-s − 3.33·11-s − 5.08·13-s + 1.62·15-s − 17-s − 5.38·19-s + 2.05·21-s − 3.60·23-s + 1.01·25-s − 3.67·27-s − 0.718·29-s + 8.53·31-s − 2.20·33-s + 7.61·35-s + 2.73·37-s − 3.36·39-s − 1.77·41-s − 10.0·43-s − 6.28·45-s − 9.54·47-s + 2.64·49-s − 0.660·51-s − 8.25·53-s − 8.19·55-s + ⋯
L(s)  = 1  + 0.381·3-s + 1.09·5-s + 1.17·7-s − 0.854·9-s − 1.00·11-s − 1.41·13-s + 0.418·15-s − 0.242·17-s − 1.23·19-s + 0.447·21-s − 0.751·23-s + 0.203·25-s − 0.707·27-s − 0.133·29-s + 1.53·31-s − 0.384·33-s + 1.28·35-s + 0.449·37-s − 0.538·39-s − 0.277·41-s − 1.52·43-s − 0.937·45-s − 1.39·47-s + 0.377·49-s − 0.0925·51-s − 1.13·53-s − 1.10·55-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 \( 1 - 0.660T + 3T^{2} \)
5 \( 1 - 2.45T + 5T^{2} \)
7 \( 1 - 3.10T + 7T^{2} \)
11 \( 1 + 3.33T + 11T^{2} \)
13 \( 1 + 5.08T + 13T^{2} \)
19 \( 1 + 5.38T + 19T^{2} \)
23 \( 1 + 3.60T + 23T^{2} \)
29 \( 1 + 0.718T + 29T^{2} \)
31 \( 1 - 8.53T + 31T^{2} \)
37 \( 1 - 2.73T + 37T^{2} \)
41 \( 1 + 1.77T + 41T^{2} \)
43 \( 1 + 10.0T + 43T^{2} \)
47 \( 1 + 9.54T + 47T^{2} \)
53 \( 1 + 8.25T + 53T^{2} \)
61 \( 1 - 0.304T + 61T^{2} \)
67 \( 1 - 2.16T + 67T^{2} \)
71 \( 1 + 15.4T + 71T^{2} \)
73 \( 1 + 16.3T + 73T^{2} \)
79 \( 1 - 14.7T + 79T^{2} \)
83 \( 1 - 9.25T + 83T^{2} \)
89 \( 1 - 1.00T + 89T^{2} \)
97 \( 1 - 11.7T + 97T^{2} \)
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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

−8.146052139277932937565797111198, −7.60109268723597148743690458222, −6.46521077200514965499338727447, −5.85517818053169678821399159737, −4.92126011780636330332783584807, −4.65055118275027703779307100849, −3.12264288376511381627467511916, −2.27403976753859290886056420512, −1.86596972900636316965011824082, 0, 1.86596972900636316965011824082, 2.27403976753859290886056420512, 3.12264288376511381627467511916, 4.65055118275027703779307100849, 4.92126011780636330332783584807, 5.85517818053169678821399159737, 6.46521077200514965499338727447, 7.60109268723597148743690458222, 8.146052139277932937565797111198

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