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  − 1.18·3-s + 1.03·5-s + 2.52·7-s − 1.58·9-s + 4.63·11-s − 5.51·13-s − 1.22·15-s − 17-s + 4.35·19-s − 3.00·21-s − 4.68·23-s − 3.93·25-s + 5.45·27-s − 8.15·29-s − 0.288·31-s − 5.49·33-s + 2.61·35-s − 8.28·37-s + 6.54·39-s + 5.92·41-s − 4.48·43-s − 1.64·45-s − 5.50·47-s − 0.615·49-s + 1.18·51-s + 3.79·53-s + 4.78·55-s + ⋯
L(s)  = 1  − 0.685·3-s + 0.462·5-s + 0.955·7-s − 0.529·9-s + 1.39·11-s − 1.52·13-s − 0.316·15-s − 0.242·17-s + 0.998·19-s − 0.654·21-s − 0.977·23-s − 0.786·25-s + 1.04·27-s − 1.51·29-s − 0.0517·31-s − 0.957·33-s + 0.441·35-s − 1.36·37-s + 1.04·39-s + 0.924·41-s − 0.683·43-s − 0.244·45-s − 0.803·47-s − 0.0878·49-s + 0.166·51-s + 0.520·53-s + 0.645·55-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 \( 1 + 1.18T + 3T^{2} \)
5 \( 1 - 1.03T + 5T^{2} \)
7 \( 1 - 2.52T + 7T^{2} \)
11 \( 1 - 4.63T + 11T^{2} \)
13 \( 1 + 5.51T + 13T^{2} \)
19 \( 1 - 4.35T + 19T^{2} \)
23 \( 1 + 4.68T + 23T^{2} \)
29 \( 1 + 8.15T + 29T^{2} \)
31 \( 1 + 0.288T + 31T^{2} \)
37 \( 1 + 8.28T + 37T^{2} \)
41 \( 1 - 5.92T + 41T^{2} \)
43 \( 1 + 4.48T + 43T^{2} \)
47 \( 1 + 5.50T + 47T^{2} \)
53 \( 1 - 3.79T + 53T^{2} \)
61 \( 1 - 0.623T + 61T^{2} \)
67 \( 1 + 10.7T + 67T^{2} \)
71 \( 1 - 1.81T + 71T^{2} \)
73 \( 1 - 0.894T + 73T^{2} \)
79 \( 1 - 0.647T + 79T^{2} \)
83 \( 1 - 5.53T + 83T^{2} \)
89 \( 1 + 1.18T + 89T^{2} \)
97 \( 1 + 9.08T + 97T^{2} \)
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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

−7.985696298128318612399314130592, −7.32507113966725405209113366857, −6.54895219532218629599173536574, −5.71486519274725610313998765120, −5.22883849352315467040763840487, −4.44477670697387273422004547474, −3.49568081282806890279949839552, −2.23775178143282389072848044736, −1.48091515573725862236542189392, 0, 1.48091515573725862236542189392, 2.23775178143282389072848044736, 3.49568081282806890279949839552, 4.44477670697387273422004547474, 5.22883849352315467040763840487, 5.71486519274725610313998765120, 6.54895219532218629599173536574, 7.32507113966725405209113366857, 7.985696298128318612399314130592

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