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  − 3-s − 3.92·5-s + 7-s − 2·9-s − 2·11-s + 4.92·13-s + 3.92·15-s + 17-s + 1.92·19-s − 21-s − 6.10·23-s + 10.3·25-s + 5·27-s − 5.92·29-s + 3.73·31-s + 2·33-s − 3.92·35-s + 8.65·37-s − 4.92·39-s + 1.92·41-s − 0.266·43-s + 7.84·45-s + 2.10·47-s − 6·49-s − 51-s + 9.37·53-s + 7.84·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.75·5-s + 0.377·7-s − 0.666·9-s − 0.603·11-s + 1.36·13-s + 1.01·15-s + 0.242·17-s + 0.440·19-s − 0.218·21-s − 1.27·23-s + 2.07·25-s + 0.962·27-s − 1.09·29-s + 0.670·31-s + 0.348·33-s − 0.662·35-s + 1.42·37-s − 0.787·39-s + 0.300·41-s − 0.0405·43-s + 1.16·45-s + 0.307·47-s − 0.857·49-s − 0.140·51-s + 1.28·53-s + 1.05·55-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 \( 1 + T + 3T^{2} \)
5 \( 1 + 3.92T + 5T^{2} \)
7 \( 1 - T + 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 - 4.92T + 13T^{2} \)
19 \( 1 - 1.92T + 19T^{2} \)
23 \( 1 + 6.10T + 23T^{2} \)
29 \( 1 + 5.92T + 29T^{2} \)
31 \( 1 - 3.73T + 31T^{2} \)
37 \( 1 - 8.65T + 37T^{2} \)
41 \( 1 - 1.92T + 41T^{2} \)
43 \( 1 + 0.266T + 43T^{2} \)
47 \( 1 - 2.10T + 47T^{2} \)
53 \( 1 - 9.37T + 53T^{2} \)
61 \( 1 + 6.92T + 61T^{2} \)
67 \( 1 - 10.2T + 67T^{2} \)
71 \( 1 - 2.37T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + 5T + 79T^{2} \)
83 \( 1 - 4.65T + 83T^{2} \)
89 \( 1 - 5.73T + 89T^{2} \)
97 \( 1 + 3.45T + 97T^{2} \)
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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

−8.029543041451572794945177254801, −7.61534479522816023643667154472, −6.56623538986048203928823539720, −5.82995478223402936263624602935, −5.08936022576443553844333274072, −4.14900301099168479685349612341, −3.62503438546142810655571113065, −2.64576878800819781500371666205, −1.05882803055996315281064102656, 0, 1.05882803055996315281064102656, 2.64576878800819781500371666205, 3.62503438546142810655571113065, 4.14900301099168479685349612341, 5.08936022576443553844333274072, 5.82995478223402936263624602935, 6.56623538986048203928823539720, 7.61534479522816023643667154472, 8.029543041451572794945177254801

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