Properties

Degree 2
Conductor $ 2^{2} \cdot 17 \cdot 59 $
Sign $-0.993 + 0.116i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.58i·3-s + 3.58i·5-s + 1.89i·7-s + 0.496·9-s − 1.56i·11-s + 4.45·13-s − 5.67·15-s + (4.09 − 0.481i)17-s − 5.20·19-s − 2.99·21-s − 3.19i·23-s − 7.84·25-s + 5.53i·27-s + 5.05i·29-s + 3.34i·31-s + ⋯
L(s)  = 1  + 0.913i·3-s + 1.60i·5-s + 0.715i·7-s + 0.165·9-s − 0.471i·11-s + 1.23·13-s − 1.46·15-s + (0.993 − 0.116i)17-s − 1.19·19-s − 0.653·21-s − 0.665i·23-s − 1.56·25-s + 1.06i·27-s + 0.938i·29-s + 0.600i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.116i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4012\)    =    \(2^{2} \cdot 17 \cdot 59\)
\( \varepsilon \)  =  $-0.993 + 0.116i$
motivic weight  =  \(1\)
character  :  $\chi_{4012} (237, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4012,\ (\ :1/2),\ -0.993 + 0.116i)$
$L(1)$  $\approx$  $1.760174722$
$L(\frac12)$  $\approx$  $1.760174722$
$L(\frac{3}{2})$   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 \( 1 + (-4.09 + 0.481i)T \)
59 \( 1 - T \)
good3 \( 1 - 1.58iT - 3T^{2} \)
5 \( 1 - 3.58iT - 5T^{2} \)
7 \( 1 - 1.89iT - 7T^{2} \)
11 \( 1 + 1.56iT - 11T^{2} \)
13 \( 1 - 4.45T + 13T^{2} \)
19 \( 1 + 5.20T + 19T^{2} \)
23 \( 1 + 3.19iT - 23T^{2} \)
29 \( 1 - 5.05iT - 29T^{2} \)
31 \( 1 - 3.34iT - 31T^{2} \)
37 \( 1 - 10.4iT - 37T^{2} \)
41 \( 1 + 9.35iT - 41T^{2} \)
43 \( 1 + 7.48T + 43T^{2} \)
47 \( 1 + 8.79T + 47T^{2} \)
53 \( 1 + 2.62T + 53T^{2} \)
61 \( 1 - 12.4iT - 61T^{2} \)
67 \( 1 - 1.37T + 67T^{2} \)
71 \( 1 - 9.38iT - 71T^{2} \)
73 \( 1 + 0.343iT - 73T^{2} \)
79 \( 1 - 0.0938iT - 79T^{2} \)
83 \( 1 + 5.61T + 83T^{2} \)
89 \( 1 - 7.62T + 89T^{2} \)
97 \( 1 + 17.3iT - 97T^{2} \)
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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

−8.665061298049885973221523540821, −8.391959908913373493013263578400, −7.16760820149429083088022725783, −6.59670001845697499212677438975, −5.93459370300242289941961754214, −5.13331659050782208334291663061, −4.07818842729413885789183366666, −3.35674273016645992748149233899, −2.85139053767975192256533064312, −1.59351717802564258831615452407, 0.52335519025261113708572628172, 1.35811224471862550452790234394, 1.97287638304118794644023227122, 3.63269050279311661805699113389, 4.25575247468047183650876227230, 5.01793120637932784377292347105, 5.98727050532105110165393863512, 6.52815332423783660642798860176, 7.57801873720087839138793526658, 7.999150434416575215080452514802

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