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  + 0.864·3-s − 1.40·5-s − 2.45·7-s − 2.25·9-s + 1.26·11-s + 3.79·13-s − 1.21·15-s − 17-s + 6.52·19-s − 2.12·21-s + 1.32·23-s − 3.02·25-s − 4.54·27-s − 7.10·29-s + 6.91·31-s + 1.09·33-s + 3.45·35-s + 6.38·37-s + 3.27·39-s − 7.21·41-s + 9.21·43-s + 3.16·45-s − 11.3·47-s − 0.958·49-s − 0.864·51-s − 12.1·53-s − 1.78·55-s + ⋯
L(s)  = 1  + 0.499·3-s − 0.628·5-s − 0.928·7-s − 0.750·9-s + 0.382·11-s + 1.05·13-s − 0.313·15-s − 0.242·17-s + 1.49·19-s − 0.463·21-s + 0.275·23-s − 0.605·25-s − 0.874·27-s − 1.31·29-s + 1.24·31-s + 0.191·33-s + 0.583·35-s + 1.04·37-s + 0.525·39-s − 1.12·41-s + 1.40·43-s + 0.471·45-s − 1.65·47-s − 0.136·49-s − 0.121·51-s − 1.67·53-s − 0.240·55-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 \( 1 - 0.864T + 3T^{2} \)
5 \( 1 + 1.40T + 5T^{2} \)
7 \( 1 + 2.45T + 7T^{2} \)
11 \( 1 - 1.26T + 11T^{2} \)
13 \( 1 - 3.79T + 13T^{2} \)
19 \( 1 - 6.52T + 19T^{2} \)
23 \( 1 - 1.32T + 23T^{2} \)
29 \( 1 + 7.10T + 29T^{2} \)
31 \( 1 - 6.91T + 31T^{2} \)
37 \( 1 - 6.38T + 37T^{2} \)
41 \( 1 + 7.21T + 41T^{2} \)
43 \( 1 - 9.21T + 43T^{2} \)
47 \( 1 + 11.3T + 47T^{2} \)
53 \( 1 + 12.1T + 53T^{2} \)
61 \( 1 + 2.31T + 61T^{2} \)
67 \( 1 + 14.7T + 67T^{2} \)
71 \( 1 + 2.56T + 71T^{2} \)
73 \( 1 - 4.85T + 73T^{2} \)
79 \( 1 - 9.22T + 79T^{2} \)
83 \( 1 + 12.6T + 83T^{2} \)
89 \( 1 - 10.0T + 89T^{2} \)
97 \( 1 + 6.67T + 97T^{2} \)
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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

−7.961394863218107692502679611125, −7.61674559906131211583254741653, −6.48014215167741697078892743149, −6.05793613825908245247721255540, −5.08556486671857651105882409824, −3.96302100652744609284563407337, −3.38358915436065618449281937130, −2.78469006648003287618148161453, −1.37948633664658586709166330200, 0, 1.37948633664658586709166330200, 2.78469006648003287618148161453, 3.38358915436065618449281937130, 3.96302100652744609284563407337, 5.08556486671857651105882409824, 6.05793613825908245247721255540, 6.48014215167741697078892743149, 7.61674559906131211583254741653, 7.961394863218107692502679611125

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