Properties

Degree 2
Conductor $ 17 \cdot 43 $
Sign $-0.637 + 0.770i$
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 + (−5.18 − 3.46i)11-s + (1.77 + 1.77i)13-s − 4.00i·16-s + (−1.05 − 3.98i)17-s + (5.18 − 7.76i)23-s + (−4.61 − 1.91i)25-s + (−6.27 + 4.19i)31-s + (−2.29 − 5.54i)36-s + (2.49 + 12.5i)41-s + (2.50 − 6.05i)43-s + (12.2 − 2.43i)44-s + (−9.65 − 9.65i)47-s + (−6.46 + 2.67i)49-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)4-s + (−0.382 + 0.923i)9-s + (−1.56 − 1.04i)11-s + (0.493 + 0.493i)13-s − 1.00i·16-s + (−0.255 − 0.966i)17-s + (1.08 − 1.61i)23-s + (−0.923 − 0.382i)25-s + (−1.12 + 0.753i)31-s + (−0.382 − 0.923i)36-s + (0.389 + 1.95i)41-s + (0.382 − 0.923i)43-s + (1.84 − 0.366i)44-s + (−1.40 − 1.40i)47-s + (−0.923 + 0.382i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.637 + 0.770i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(731\)    =    \(17 \cdot 43\)
\( \varepsilon \)  =  $-0.637 + 0.770i$
motivic weight  =  \(1\)
character  :  $\chi_{731} (515, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 731,\ (\ :1/2),\ -0.637 + 0.770i)$
$L(1)$  $\approx$  $0.106293 - 0.225895i$
$L(\frac12)$  $\approx$  $0.106293 - 0.225895i$
$L(\frac{3}{2})$   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 \( 1 + (1.05 + 3.98i)T \)
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 \( 1 + (5.18 + 3.46i)T + (4.20 + 10.1i)T^{2} \)
13 \( 1 + (-1.77 - 1.77i)T + 13iT^{2} \)
19 \( 1 + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-5.18 + 7.76i)T + (-8.80 - 21.2i)T^{2} \)
29 \( 1 + (-26.7 - 11.0i)T^{2} \)
31 \( 1 + (6.27 - 4.19i)T + (11.8 - 28.6i)T^{2} \)
37 \( 1 + (14.1 - 34.1i)T^{2} \)
41 \( 1 + (-2.49 - 12.5i)T + (-37.8 + 15.6i)T^{2} \)
47 \( 1 + (9.65 + 9.65i)T + 47iT^{2} \)
53 \( 1 + (5.55 + 13.4i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (2.19 + 0.907i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (-56.3 + 23.3i)T^{2} \)
67 \( 1 - 0.318iT - 67T^{2} \)
71 \( 1 + (27.1 - 65.5i)T^{2} \)
73 \( 1 + (67.4 + 27.9i)T^{2} \)
79 \( 1 + (-2.23 - 1.49i)T + (30.2 + 72.9i)T^{2} \)
83 \( 1 + (6.05 - 2.50i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + 89iT^{2} \)
97 \( 1 + (9.90 + 1.96i)T + (89.6 + 37.1i)T^{2} \)
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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

−10.11640718711456479170856355678, −9.001179274367136507546393392968, −8.309752351971606787516974653436, −7.79673089505412598802858097951, −6.61808590076380342954414974818, −5.27053897384241798980965972861, −4.78112713590824735092975668925, −3.36534670356337475727394104588, −2.49484570194487245473473578513, −0.12738536170039738841728891413, 1.63879845114971410305473469140, 3.24972002939365257576016959410, 4.36043033490189218450415857789, 5.47944474766573794351233762519, 5.95304780087840629401449470459, 7.33290339183507229806892355604, 8.135199108470660968959472786409, 9.248242876924729168401417144806, 9.676420824830287195912374557545, 10.68351815724866780521749302897

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