Properties

Degree 2
Conductor $ 17 \cdot 43 $
Sign $0.489 + 0.871i$
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 + (2.75 − 0.548i)11-s + (1.77 + 1.77i)13-s − 4.00i·16-s + (3.98 − 1.05i)17-s + (−0.988 − 4.96i)23-s + (1.91 − 4.61i)25-s + (−0.443 − 0.0882i)31-s + (−5.54 + 2.29i)36-s + (−7.90 − 5.28i)41-s + (6.05 + 2.50i)43-s + (3.12 − 4.67i)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 + (0.831 − 0.165i)11-s + (0.493 + 0.493i)13-s − 1.00i·16-s + (0.966 − 0.255i)17-s + (−0.206 − 1.03i)23-s + (0.382 − 0.923i)25-s + (−0.0796 − 0.0158i)31-s + (−0.923 + 0.382i)36-s + (−1.23 − 0.825i)41-s + (0.923 + 0.382i)43-s + (0.470 − 0.704i)44-s + (−0.136 − 0.136i)47-s + (0.382 + 0.923i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.489 + 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.489 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(731\)    =    \(17 \cdot 43\)
\( \varepsilon \)  =  $0.489 + 0.871i$
motivic weight  =  \(1\)
character  :  $\chi_{731} (558, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 731,\ (\ :1/2),\ 0.489 + 0.871i)$
$L(1)$  $\approx$  $1.51830 - 0.888635i$
$L(\frac12)$  $\approx$  $1.51830 - 0.888635i$
$L(\frac{3}{2})$   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 \( 1 + (-2.75 + 0.548i)T + (10.1 - 4.20i)T^{2} \)
13 \( 1 + (-1.77 - 1.77i)T + 13iT^{2} \)
19 \( 1 + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (0.988 + 4.96i)T + (-21.2 + 8.80i)T^{2} \)
29 \( 1 + (11.0 - 26.7i)T^{2} \)
31 \( 1 + (0.443 + 0.0882i)T + (28.6 + 11.8i)T^{2} \)
37 \( 1 + (34.1 + 14.1i)T^{2} \)
41 \( 1 + (7.90 + 5.28i)T + (15.6 + 37.8i)T^{2} \)
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 \( 1 + (-14.0 + 2.79i)T + (72.9 - 30.2i)T^{2} \)
83 \( 1 + (-2.50 - 6.05i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + 89iT^{2} \)
97 \( 1 + (2.67 + 4.00i)T + (-37.1 + 89.6i)T^{2} \)
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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

−10.39091736782749225440929869227, −9.353266359579751199295770700373, −8.687109857957817285239969629355, −7.56485929286055877064538131204, −6.41058789045243355006136061590, −6.09561625428432192347132114399, −4.91780511139432617750338077035, −3.57880865783932753836423465068, −2.42621406795068358130534023001, −0.977055366957075231561650685697, 1.64158720805134176462891535082, 3.06454655998421704355070936476, 3.74192659700056524068674739800, 5.29441855617555850804954481638, 6.13434074997621124870335806169, 7.12840364266308489868556510799, 7.966736497581805911881530317475, 8.636484669217394581073020271609, 9.662299133182291221919492817411, 10.73909340529681271490567570888

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