Properties

Degree 2
Conductor $ 17 \cdot 43 $
Sign $0.314 + 0.949i$
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 + (−0.955 − 4.80i)11-s + (−4.77 − 4.77i)13-s − 4.00i·16-s + (2.07 + 3.56i)17-s + (5.47 − 1.08i)23-s + (−1.91 + 4.61i)25-s + (0.749 − 3.76i)31-s + (5.54 − 2.29i)36-s + (−2.38 + 3.56i)41-s + (6.05 + 2.50i)43-s + (−8.14 − 5.44i)44-s + (−6.16 − 6.16i)47-s + (−2.67 − 6.46i)49-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)4-s + (0.923 + 0.382i)9-s + (−0.288 − 1.44i)11-s + (−1.32 − 1.32i)13-s − 1.00i·16-s + (0.502 + 0.864i)17-s + (1.14 − 0.227i)23-s + (−0.382 + 0.923i)25-s + (0.134 − 0.676i)31-s + (0.923 − 0.382i)36-s + (−0.371 + 0.556i)41-s + (0.923 + 0.382i)43-s + (−1.22 − 0.820i)44-s + (−0.898 − 0.898i)47-s + (−0.382 − 0.923i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.314 + 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.314 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(731\)    =    \(17 \cdot 43\)
\( \varepsilon \)  =  $0.314 + 0.949i$
motivic weight  =  \(1\)
character  :  $\chi_{731} (343, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 731,\ (\ :1/2),\ 0.314 + 0.949i)$
$L(1)$  $\approx$  $1.41590 - 1.02221i$
$L(\frac12)$  $\approx$  $1.41590 - 1.02221i$
$L(\frac{3}{2})$   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 \( 1 + (-2.07 - 3.56i)T \)
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 \( 1 + (0.955 + 4.80i)T + (-10.1 + 4.20i)T^{2} \)
13 \( 1 + (4.77 + 4.77i)T + 13iT^{2} \)
19 \( 1 + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (-5.47 + 1.08i)T + (21.2 - 8.80i)T^{2} \)
29 \( 1 + (-11.0 + 26.7i)T^{2} \)
31 \( 1 + (-0.749 + 3.76i)T + (-28.6 - 11.8i)T^{2} \)
37 \( 1 + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (2.38 - 3.56i)T + (-15.6 - 37.8i)T^{2} \)
47 \( 1 + (6.16 + 6.16i)T + 47iT^{2} \)
53 \( 1 + (-10.1 + 4.22i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-3.46 + 8.36i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-23.3 - 56.3i)T^{2} \)
67 \( 1 - 11.3iT - 67T^{2} \)
71 \( 1 + (-65.5 - 27.1i)T^{2} \)
73 \( 1 + (27.9 - 67.4i)T^{2} \)
79 \( 1 + (-2.96 - 14.9i)T + (-72.9 + 30.2i)T^{2} \)
83 \( 1 + (-2.50 - 6.05i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + 89iT^{2} \)
97 \( 1 + (16.2 - 10.8i)T + (37.1 - 89.6i)T^{2} \)
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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

−10.23187490053831188709827379289, −9.708841875866702527877146475194, −8.330310116900740138981184434391, −7.58423323694122807828596365833, −6.73304007561082408087433347890, −5.60048506729065633123138691537, −5.11495284957919129656741418678, −3.48821797523684966364467641592, −2.39879415646494346275757477088, −0.926348790280399283107982104649, 1.80988955943073015625644373954, 2.80021182392715715590433274685, 4.22055936726821987043238808044, 4.88051286791954819606848631794, 6.50390540353131203251897460597, 7.36571372225256836925986243792, 7.41121384406685907807796203257, 9.003579524017565967058285029427, 9.713019219170829794839425870387, 10.45507773724410742443829885261

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