Properties

Degree 2
Conductor $ 17 \cdot 43 $
Sign $-0.711 - 0.703i$
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 + (−4.38 − 0.872i)11-s + (−4.77 + 4.77i)13-s + 4.00i·16-s + (−2.07 + 3.56i)17-s + (1.52 − 7.64i)23-s + (1.91 + 4.61i)25-s + (10.2 − 2.03i)31-s + (−5.54 − 2.29i)36-s + (−10.0 + 6.70i)41-s + (−6.05 + 2.50i)43-s + (−4.97 − 7.44i)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 + (−1.32 − 0.263i)11-s + (−1.32 + 1.32i)13-s + 1.00i·16-s + (−0.502 + 0.864i)17-s + (0.317 − 1.59i)23-s + (0.382 + 0.923i)25-s + (1.84 − 0.366i)31-s + (−0.923 − 0.382i)36-s + (−1.56 + 1.04i)41-s + (−0.923 + 0.382i)43-s + (−0.749 − 1.12i)44-s + (0.898 − 0.898i)47-s + (0.382 − 0.923i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 - 0.703i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.711 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(731\)    =    \(17 \cdot 43\)
\( \varepsilon \)  =  $-0.711 - 0.703i$
motivic weight  =  \(1\)
character  :  $\chi_{731} (300, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 731,\ (\ :1/2),\ -0.711 - 0.703i)$
$L(1)$  $\approx$  $0.367470 + 0.894426i$
$L(\frac12)$  $\approx$  $0.367470 + 0.894426i$
$L(\frac{3}{2})$   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 \( 1 + (4.38 + 0.872i)T + (10.1 + 4.20i)T^{2} \)
13 \( 1 + (4.77 - 4.77i)T - 13iT^{2} \)
19 \( 1 + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-1.52 + 7.64i)T + (-21.2 - 8.80i)T^{2} \)
29 \( 1 + (11.0 + 26.7i)T^{2} \)
31 \( 1 + (-10.2 + 2.03i)T + (28.6 - 11.8i)T^{2} \)
37 \( 1 + (34.1 - 14.1i)T^{2} \)
41 \( 1 + (10.0 - 6.70i)T + (15.6 - 37.8i)T^{2} \)
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 \( 1 + (-9.03 - 1.79i)T + (72.9 + 30.2i)T^{2} \)
83 \( 1 + (2.50 - 6.05i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 - 89iT^{2} \)
97 \( 1 + (-1.58 + 2.37i)T + (-37.1 - 89.6i)T^{2} \)
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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

−10.73580007837636831079779815375, −10.07269735903560581797440595277, −8.608800791518867945384301759302, −8.303506466371751615017087613733, −7.17532130982921236183237540132, −6.51643652807600702783285577667, −5.30108663664572023393021741098, −4.32108625254821952779104924156, −2.83027011566362021345788607150, −2.27237243837536742143792983319, 0.44033068918331144713624285261, 2.43579154902726021476492979937, 3.01721708468101531013998040064, 5.09957847238062074751101362596, 5.32937969484098975246824400751, 6.55093474340671184324090532934, 7.44295563965590973947249908106, 8.198706890359962021276411764356, 9.427578834247746463050709065274, 10.21016670825225218775024689672

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