Properties

Degree 2
Conductor $ 2^{2} \cdot 17 \cdot 59 $
Sign $-0.252 + 0.967i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.13i·3-s − 2.09i·5-s + 2.71i·7-s + 1.71·9-s − 3.86i·11-s − 2.19·13-s − 2.37·15-s + (1.04 − 3.98i)17-s + 7.57·19-s + 3.07·21-s + 3.94i·23-s + 0.603·25-s − 5.34i·27-s − 2.64i·29-s − 4.90i·31-s + ⋯
L(s)  = 1  − 0.654i·3-s − 0.937i·5-s + 1.02i·7-s + 0.572·9-s − 1.16i·11-s − 0.608·13-s − 0.613·15-s + (0.252 − 0.967i)17-s + 1.73·19-s + 0.669·21-s + 0.821i·23-s + 0.120·25-s − 1.02i·27-s − 0.491i·29-s − 0.880i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4012\)    =    \(2^{2} \cdot 17 \cdot 59\)
\( \varepsilon \)  =  $-0.252 + 0.967i$
motivic weight  =  \(1\)
character  :  $\chi_{4012} (237, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4012,\ (\ :1/2),\ -0.252 + 0.967i)$
$L(1)$  $\approx$  $2.009140297$
$L(\frac12)$  $\approx$  $2.009140297$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;17,\;59\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;17,\;59\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 + (-1.04 + 3.98i)T \)
59 \( 1 - T \)
good3 \( 1 + 1.13iT - 3T^{2} \)
5 \( 1 + 2.09iT - 5T^{2} \)
7 \( 1 - 2.71iT - 7T^{2} \)
11 \( 1 + 3.86iT - 11T^{2} \)
13 \( 1 + 2.19T + 13T^{2} \)
19 \( 1 - 7.57T + 19T^{2} \)
23 \( 1 - 3.94iT - 23T^{2} \)
29 \( 1 + 2.64iT - 29T^{2} \)
31 \( 1 + 4.90iT - 31T^{2} \)
37 \( 1 - 11.3iT - 37T^{2} \)
41 \( 1 - 4.94iT - 41T^{2} \)
43 \( 1 + 5.42T + 43T^{2} \)
47 \( 1 - 6.03T + 47T^{2} \)
53 \( 1 + 1.40T + 53T^{2} \)
61 \( 1 + 13.3iT - 61T^{2} \)
67 \( 1 - 1.98T + 67T^{2} \)
71 \( 1 + 4.42iT - 71T^{2} \)
73 \( 1 - 1.70iT - 73T^{2} \)
79 \( 1 + 4.77iT - 79T^{2} \)
83 \( 1 - 9.20T + 83T^{2} \)
89 \( 1 + 4.58T + 89T^{2} \)
97 \( 1 + 14.2iT - 97T^{2} \)
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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

−8.077931688518768798362721024609, −7.70575552100011916706766323133, −6.78551015304130667063886059812, −5.93694330759916094106748546129, −5.24422990152635717090515661439, −4.73400696170491132636015235874, −3.40831046707652419700677490260, −2.64886719746522558069144025008, −1.48176028474446509081075650660, −0.66488539385863603208511549114, 1.18608429562315960849453801644, 2.36153618202169901682200463949, 3.43947579995335641400045544121, 4.01795765514467711322402531760, 4.78735991495933105710136106719, 5.55425537282578870729775927197, 6.79336922264861429409606452610, 7.22023709919276602965279302262, 7.55400846731536598822161834050, 8.779357859409272993904401895539

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