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  − 2.36·3-s + 2.61·5-s − 2.25·7-s + 2.61·9-s − 6.06·11-s + 5.42·13-s − 6.19·15-s − 17-s − 4.71·19-s + 5.34·21-s + 9.23·23-s + 1.82·25-s + 0.917·27-s + 4.85·29-s − 0.261·31-s + 14.3·33-s − 5.89·35-s + 4.91·37-s − 12.8·39-s − 5.42·41-s − 5.25·43-s + 6.82·45-s − 1.34·47-s − 1.91·49-s + 2.36·51-s + 5.39·53-s − 15.8·55-s + ⋯
L(s)  = 1  − 1.36·3-s + 1.16·5-s − 0.852·7-s + 0.870·9-s − 1.82·11-s + 1.50·13-s − 1.59·15-s − 0.242·17-s − 1.08·19-s + 1.16·21-s + 1.92·23-s + 0.365·25-s + 0.176·27-s + 0.901·29-s − 0.0469·31-s + 2.49·33-s − 0.996·35-s + 0.807·37-s − 2.05·39-s − 0.848·41-s − 0.801·43-s + 1.01·45-s − 0.195·47-s − 0.273·49-s + 0.331·51-s + 0.740·53-s − 2.13·55-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 \( 1 + 2.36T + 3T^{2} \)
5 \( 1 - 2.61T + 5T^{2} \)
7 \( 1 + 2.25T + 7T^{2} \)
11 \( 1 + 6.06T + 11T^{2} \)
13 \( 1 - 5.42T + 13T^{2} \)
19 \( 1 + 4.71T + 19T^{2} \)
23 \( 1 - 9.23T + 23T^{2} \)
29 \( 1 - 4.85T + 29T^{2} \)
31 \( 1 + 0.261T + 31T^{2} \)
37 \( 1 - 4.91T + 37T^{2} \)
41 \( 1 + 5.42T + 41T^{2} \)
43 \( 1 + 5.25T + 43T^{2} \)
47 \( 1 + 1.34T + 47T^{2} \)
53 \( 1 - 5.39T + 53T^{2} \)
61 \( 1 + 12.6T + 61T^{2} \)
67 \( 1 + 3.44T + 67T^{2} \)
71 \( 1 + 2.07T + 71T^{2} \)
73 \( 1 + 0.551T + 73T^{2} \)
79 \( 1 - 15.1T + 79T^{2} \)
83 \( 1 - 7.01T + 83T^{2} \)
89 \( 1 + 8.76T + 89T^{2} \)
97 \( 1 + 1.65T + 97T^{2} \)
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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

−8.162934404900803986723478935259, −6.95587378900937097733713860403, −6.37280630883751308291425553713, −5.95003682724910985638754628234, −5.23843154731192615724076653263, −4.66258928304263679301053232640, −3.27860752450531131492448594083, −2.46799208411170099191147324624, −1.20268058637710312574565973240, 0, 1.20268058637710312574565973240, 2.46799208411170099191147324624, 3.27860752450531131492448594083, 4.66258928304263679301053232640, 5.23843154731192615724076653263, 5.95003682724910985638754628234, 6.37280630883751308291425553713, 6.95587378900937097733713860403, 8.162934404900803986723478935259

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