Properties

Degree 2
Conductor $ 17 \cdot 43 $
Sign $0.694 + 0.719i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.41 + 1.41i)4-s + (1.14 − 2.77i)9-s + (−0.165 + 0.248i)11-s + (−4.77 − 4.77i)13-s − 4.00i·16-s + (3.56 − 2.07i)17-s + (7.86 + 5.25i)23-s + (4.61 + 1.91i)25-s + (−2.59 − 3.88i)31-s + (2.29 + 5.54i)36-s + (11.3 − 2.25i)41-s + (2.50 − 6.05i)43-s + (−0.116 − 0.585i)44-s + (−7.48 − 7.48i)47-s + (6.46 − 2.67i)49-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)4-s + (0.382 − 0.923i)9-s + (−0.0500 + 0.0748i)11-s + (−1.32 − 1.32i)13-s − 1.00i·16-s + (0.864 − 0.502i)17-s + (1.64 + 1.09i)23-s + (0.923 + 0.382i)25-s + (−0.466 − 0.698i)31-s + (0.382 + 0.923i)36-s + (1.77 − 0.352i)41-s + (0.382 − 0.923i)43-s + (−0.0175 − 0.0882i)44-s + (−1.09 − 1.09i)47-s + (0.923 − 0.382i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.694 + 0.719i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.694 + 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(731\)    =    \(17 \cdot 43\)
\( \varepsilon \)  =  $0.694 + 0.719i$
motivic weight  =  \(1\)
character  :  $\chi_{731} (386, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 731,\ (\ :1/2),\ 0.694 + 0.719i)$
$L(1)$  $\approx$  $1.05492 - 0.448288i$
$L(\frac12)$  $\approx$  $1.05492 - 0.448288i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{17,\;43\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{17,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad17 \( 1 + (-3.56 + 2.07i)T \)
43 \( 1 + (-2.50 + 6.05i)T \)
good2 \( 1 + (1.41 - 1.41i)T^{2} \)
3 \( 1 + (-1.14 + 2.77i)T^{2} \)
5 \( 1 + (-4.61 - 1.91i)T^{2} \)
7 \( 1 + (-6.46 + 2.67i)T^{2} \)
11 \( 1 + (0.165 - 0.248i)T + (-4.20 - 10.1i)T^{2} \)
13 \( 1 + (4.77 + 4.77i)T + 13iT^{2} \)
19 \( 1 + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-7.86 - 5.25i)T + (8.80 + 21.2i)T^{2} \)
29 \( 1 + (26.7 + 11.0i)T^{2} \)
31 \( 1 + (2.59 + 3.88i)T + (-11.8 + 28.6i)T^{2} \)
37 \( 1 + (-14.1 + 34.1i)T^{2} \)
41 \( 1 + (-11.3 + 2.25i)T + (37.8 - 15.6i)T^{2} \)
47 \( 1 + (7.48 + 7.48i)T + 47iT^{2} \)
53 \( 1 + (3.63 + 8.77i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (11.4 + 4.74i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (56.3 - 23.3i)T^{2} \)
67 \( 1 - 11.7iT - 67T^{2} \)
71 \( 1 + (-27.1 + 65.5i)T^{2} \)
73 \( 1 + (-67.4 - 27.9i)T^{2} \)
79 \( 1 + (2.35 - 3.52i)T + (-30.2 - 72.9i)T^{2} \)
83 \( 1 + (6.05 - 2.50i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + 89iT^{2} \)
97 \( 1 + (-3.08 + 15.4i)T + (-89.6 - 37.1i)T^{2} \)
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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

−9.976595024605230395088842341734, −9.482892904141326702887891895977, −8.655120674170661225252248172511, −7.45703612907077964768930388055, −7.22316821584158976848051637927, −5.55336523677684076654087311744, −4.88685647037772670335126771386, −3.60900474421162264325293355992, −2.87337444415940989826126766604, −0.68473308982348851222084806348, 1.34846778454871446358978195154, 2.74155726713514193213384104394, 4.58885791501115744027241878651, 4.71651165442338859610186318685, 5.99780122893530728475287517443, 7.02433439760827122323566329951, 7.910540378873887359462850732890, 9.046818762365830417204830062081, 9.524762696326202975567463586173, 10.59232351734711548310328961543

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