Properties

Degree 2
Conductor $ 17 \cdot 43 $
Sign $0.998 - 0.0522i$
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 + (2.77 + 1.14i)9-s + (1.17 + 5.89i)11-s + (1.77 + 1.77i)13-s − 4.00i·16-s + (−3.98 + 1.05i)17-s + (7.98 − 1.58i)23-s + (−1.91 + 4.61i)25-s + (2.17 − 10.9i)31-s + (5.54 − 2.29i)36-s + (4.76 − 7.13i)41-s + (−6.05 − 2.50i)43-s + (9.99 + 6.67i)44-s + (0.935 + 0.935i)47-s + (−2.67 − 6.46i)49-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)4-s + (0.923 + 0.382i)9-s + (0.353 + 1.77i)11-s + (0.493 + 0.493i)13-s − 1.00i·16-s + (−0.966 + 0.255i)17-s + (1.66 − 0.331i)23-s + (−0.382 + 0.923i)25-s + (0.389 − 1.95i)31-s + (0.923 − 0.382i)36-s + (0.744 − 1.11i)41-s + (−0.923 − 0.382i)43-s + (1.50 + 1.00i)44-s + (0.136 + 0.136i)47-s + (−0.382 − 0.923i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0522i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(731\)    =    \(17 \cdot 43\)
\( \varepsilon \)  =  $0.998 - 0.0522i$
motivic weight  =  \(1\)
character  :  $\chi_{731} (343, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 731,\ (\ :1/2),\ 0.998 - 0.0522i)$
$L(1)$  $\approx$  $1.95880 + 0.0512224i$
$L(\frac12)$  $\approx$  $1.95880 + 0.0512224i$
$L(\frac{3}{2})$   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 \( 1 + (3.98 - 1.05i)T \)
43 \( 1 + (6.05 + 2.50i)T \)
good2 \( 1 + (-1.41 + 1.41i)T^{2} \)
3 \( 1 + (-2.77 - 1.14i)T^{2} \)
5 \( 1 + (1.91 - 4.61i)T^{2} \)
7 \( 1 + (2.67 + 6.46i)T^{2} \)
11 \( 1 + (-1.17 - 5.89i)T + (-10.1 + 4.20i)T^{2} \)
13 \( 1 + (-1.77 - 1.77i)T + 13iT^{2} \)
19 \( 1 + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (-7.98 + 1.58i)T + (21.2 - 8.80i)T^{2} \)
29 \( 1 + (-11.0 + 26.7i)T^{2} \)
31 \( 1 + (-2.17 + 10.9i)T + (-28.6 - 11.8i)T^{2} \)
37 \( 1 + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (-4.76 + 7.13i)T + (-15.6 - 37.8i)T^{2} \)
47 \( 1 + (-0.935 - 0.935i)T + 47iT^{2} \)
53 \( 1 + (1.00 - 0.414i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (5.80 - 14.0i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-23.3 - 56.3i)T^{2} \)
67 \( 1 - 16.3iT - 67T^{2} \)
71 \( 1 + (-65.5 - 27.1i)T^{2} \)
73 \( 1 + (27.9 - 67.4i)T^{2} \)
79 \( 1 + (2.05 + 10.3i)T + (-72.9 + 30.2i)T^{2} \)
83 \( 1 + (2.50 + 6.05i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + 89iT^{2} \)
97 \( 1 + (-15.8 + 10.6i)T + (37.1 - 89.6i)T^{2} \)
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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

−10.35274645361097507612404909008, −9.679047284035662707445223453825, −8.907333745510485612077318532157, −7.28597924302888230546883221320, −7.13786854223794080689356397927, −6.08289668576294805050362058196, −4.87143150160277181182949407965, −4.13362769801832389696259986490, −2.33414469757957925729261403225, −1.50637238504644656825039950127, 1.22028323507647595613900731734, 2.93660396385215334960513066360, 3.60420476266569301259935374081, 4.87470092691964118422788739716, 6.35319261584338411906399166128, 6.65780239566044025192647334746, 7.87213751751247918140183120244, 8.570058832255667602169144734663, 9.387494257156118219730139786179, 10.77048161184473589099307608225

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