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  :  $\chi_{731} (300, \cdot )$
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

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

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