Properties

Degree 2
Conductor $ 17 \cdot 43 $
Sign $-0.891 + 0.453i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.473 − 2.07i)2-s + (−3.20 − 0.730i)3-s + (−2.28 + 1.09i)4-s + (−0.606 + 0.483i)5-s + 6.98i·6-s + 4.01i·7-s + (0.704 + 0.883i)8-s + (7.01 + 3.37i)9-s + (1.29 + 1.02i)10-s + (−0.0655 + 0.136i)11-s + (8.10 − 1.84i)12-s + (−2.82 − 3.53i)13-s + (8.33 − 1.90i)14-s + (2.29 − 1.10i)15-s + (−1.65 + 2.07i)16-s + (−3.20 + 2.59i)17-s + ⋯
L(s)  = 1  + (−0.334 − 1.46i)2-s + (−1.84 − 0.421i)3-s + (−1.14 + 0.549i)4-s + (−0.271 + 0.216i)5-s + 2.85i·6-s + 1.51i·7-s + (0.249 + 0.312i)8-s + (2.33 + 1.12i)9-s + (0.408 + 0.325i)10-s + (−0.0197 + 0.0410i)11-s + (2.33 − 0.533i)12-s + (−0.782 − 0.981i)13-s + (2.22 − 0.508i)14-s + (0.592 − 0.285i)15-s + (−0.414 + 0.519i)16-s + (−0.777 + 0.629i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 + 0.453i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(731\)    =    \(17 \cdot 43\)
\( \varepsilon \)  =  $-0.891 + 0.453i$
motivic weight  =  \(1\)
character  :  $\chi_{731} (16, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 731,\ (\ :1/2),\ -0.891 + 0.453i)$
$L(1)$  $\approx$  $0.0963633 - 0.401644i$
$L(\frac12)$  $\approx$  $0.0963633 - 0.401644i$
$L(\frac{3}{2})$   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 \( 1 + (3.20 - 2.59i)T \)
43 \( 1 + (5.20 - 3.99i)T \)
good2 \( 1 + (0.473 + 2.07i)T + (-1.80 + 0.867i)T^{2} \)
3 \( 1 + (3.20 + 0.730i)T + (2.70 + 1.30i)T^{2} \)
5 \( 1 + (0.606 - 0.483i)T + (1.11 - 4.87i)T^{2} \)
7 \( 1 - 4.01iT - 7T^{2} \)
11 \( 1 + (0.0655 - 0.136i)T + (-6.85 - 8.60i)T^{2} \)
13 \( 1 + (2.82 + 3.53i)T + (-2.89 + 12.6i)T^{2} \)
19 \( 1 + (-5.48 + 2.64i)T + (11.8 - 14.8i)T^{2} \)
23 \( 1 + (-2.64 + 5.48i)T + (-14.3 - 17.9i)T^{2} \)
29 \( 1 + (3.86 - 0.881i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (-5.46 + 1.24i)T + (27.9 - 13.4i)T^{2} \)
37 \( 1 - 1.77iT - 37T^{2} \)
41 \( 1 + (-0.498 + 0.113i)T + (36.9 - 17.7i)T^{2} \)
47 \( 1 + (-6.46 + 3.11i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-3.96 + 4.97i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (-8.70 + 10.9i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (6.39 + 1.45i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + (7.90 - 3.80i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (3.49 + 7.25i)T + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (-0.479 + 0.382i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 - 5.60iT - 79T^{2} \)
83 \( 1 + (-2.74 + 12.0i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (0.803 - 3.51i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + (-7.83 + 16.2i)T + (-60.4 - 75.8i)T^{2} \)
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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

−10.32195810132217226317829102170, −9.622022007614325256158010650628, −8.562927686052055849212310231797, −7.30464635675702561615815411934, −6.31520215797627866375854687088, −5.41706947224731020422508795163, −4.65100336238805897151795576031, −3.01415013767785309112790687120, −1.91501592952053169397051438206, −0.44622582536985333841075531887, 0.851704522977709713226561418822, 4.06606870947595957831952766216, 4.73033670114561067339872237798, 5.51717656241158176674732134874, 6.47175280767652798995106995987, 7.25032790856411993986030183519, 7.49939279159054271691287560600, 9.151427891259083046592791027369, 9.863172594860691815536664912743, 10.62726617608932142873534888347

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