Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 59 $
Sign $0.867 + 0.497i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + (−1.43 − 0.975i)3-s + 4-s + 0.0999i·5-s + (1.43 + 0.975i)6-s + 2.48·7-s − 8-s + (1.09 + 2.79i)9-s − 0.0999i·10-s − 1.50·11-s + (−1.43 − 0.975i)12-s + 3.74i·13-s − 2.48·14-s + (0.0974 − 0.143i)15-s + 16-s − 6.68i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.826 − 0.563i)3-s + 0.5·4-s + 0.0446i·5-s + (0.584 + 0.398i)6-s + 0.939·7-s − 0.353·8-s + (0.365 + 0.930i)9-s − 0.0315i·10-s − 0.453·11-s + (−0.413 − 0.281i)12-s + 1.03i·13-s − 0.664·14-s + (0.0251 − 0.0369i)15-s + 0.250·16-s − 1.62i·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 + 0.497i)\, \overline{\Lambda}(2-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.867 + 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\n\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(354\)    =    \(2 \cdot 3 \cdot 59\)
\( \varepsilon \)  =  $0.867 + 0.497i$
motivic weight  =  \(1\)
character  :  $\chi_{354} (353, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 354,\ (\ :1/2),\ 0.867 + 0.497i)$
$L(1)$  $\approx$  $0.800002 - 0.212881i$
$L(\frac12)$  $\approx$  $0.800002 - 0.212881i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;59\}$, \(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;59\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (1.43 + 0.975i)T \)
59 \( 1 + (-7.65 + 0.598i)T \)
good5 \( 1 - 0.0999iT - 5T^{2} \)
7 \( 1 - 2.48T + 7T^{2} \)
11 \( 1 + 1.50T + 11T^{2} \)
13 \( 1 - 3.74iT - 13T^{2} \)
17 \( 1 + 6.68iT - 17T^{2} \)
19 \( 1 - 3.99T + 19T^{2} \)
23 \( 1 - 7.31T + 23T^{2} \)
29 \( 1 - 3.18iT - 29T^{2} \)
31 \( 1 + 9.23iT - 31T^{2} \)
37 \( 1 - 0.665iT - 37T^{2} \)
41 \( 1 - 3.99iT - 41T^{2} \)
43 \( 1 + 10.0iT - 43T^{2} \)
47 \( 1 - 4.30T + 47T^{2} \)
53 \( 1 - 0.250iT - 53T^{2} \)
61 \( 1 - 13.2iT - 61T^{2} \)
67 \( 1 - 7.06iT - 67T^{2} \)
71 \( 1 - 10.1iT - 71T^{2} \)
73 \( 1 + 11.3iT - 73T^{2} \)
79 \( 1 + 9.13T + 79T^{2} \)
83 \( 1 - 5.43T + 83T^{2} \)
89 \( 1 + 8.23T + 89T^{2} \)
97 \( 1 + 3.83iT - 97T^{2} \)
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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

−11.48107862452705285700905108642, −10.67772622660984370747177048299, −9.536876475017593000485888984861, −8.578342082048149276589812942227, −7.32849414381893626722067905846, −7.04000358382428155309849432189, −5.52840726495451691589822231831, −4.71684794415080759783679346140, −2.53914572379603289372895817358, −1.05572735147348133659319009690, 1.18898358190915617803119260905, 3.23681724461727463686502788979, 4.82565996842722380661565077074, 5.59807638040890677414896667440, 6.80981651484537241887001326419, 7.939178463539502365175350072294, 8.758178959144775932034239080901, 9.893224881482748017396226711410, 10.80282297292399115717228291231, 11.04880464750286738869841662514

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