Properties

Degree 2
Conductor $ 17 \cdot 43 $
Sign $0.650 - 0.759i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.69 + 2.12i)2-s + (−0.339 − 2.25i)3-s + (−1.20 − 5.26i)4-s + (0.0789 + 0.115i)5-s + (5.37 + 3.10i)6-s + (−0.309 + 0.178i)7-s + (8.33 + 4.01i)8-s + (−2.10 + 0.648i)9-s + (−0.380 − 0.0285i)10-s + (−3.60 − 0.823i)11-s + (−11.4 + 4.49i)12-s + (0.376 + 5.02i)13-s + (0.144 − 0.959i)14-s + (0.234 − 0.217i)15-s + (−12.9 + 6.23i)16-s + (−0.731 + 4.05i)17-s + ⋯
L(s)  = 1  + (−1.19 + 1.50i)2-s + (−0.196 − 1.30i)3-s + (−0.600 − 2.63i)4-s + (0.0353 + 0.0518i)5-s + (2.19 + 1.26i)6-s + (−0.116 + 0.0674i)7-s + (2.94 + 1.41i)8-s + (−0.700 + 0.216i)9-s + (−0.120 − 0.00901i)10-s + (−1.08 − 0.248i)11-s + (−3.30 + 1.29i)12-s + (0.104 + 1.39i)13-s + (0.0386 − 0.256i)14-s + (0.0605 − 0.0561i)15-s + (−3.23 + 1.55i)16-s + (−0.177 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.650 - 0.759i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.650 - 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(731\)    =    \(17 \cdot 43\)
\( \varepsilon \)  =  $0.650 - 0.759i$
motivic weight  =  \(1\)
character  :  $\chi_{731} (67, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 731,\ (\ :1/2),\ 0.650 - 0.759i)$
$L(1)$  $\approx$  $0.589453 + 0.271203i$
$L(\frac12)$  $\approx$  $0.589453 + 0.271203i$
$L(\frac{3}{2})$   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 \( 1 + (0.731 - 4.05i)T \)
43 \( 1 + (-6.39 - 1.46i)T \)
good2 \( 1 + (1.69 - 2.12i)T + (-0.445 - 1.94i)T^{2} \)
3 \( 1 + (0.339 + 2.25i)T + (-2.86 + 0.884i)T^{2} \)
5 \( 1 + (-0.0789 - 0.115i)T + (-1.82 + 4.65i)T^{2} \)
7 \( 1 + (0.309 - 0.178i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.60 + 0.823i)T + (9.91 + 4.77i)T^{2} \)
13 \( 1 + (-0.376 - 5.02i)T + (-12.8 + 1.93i)T^{2} \)
19 \( 1 + (-6.96 - 2.14i)T + (15.6 + 10.7i)T^{2} \)
23 \( 1 + (4.15 - 4.47i)T + (-1.71 - 22.9i)T^{2} \)
29 \( 1 + (-0.915 + 6.07i)T + (-27.7 - 8.54i)T^{2} \)
31 \( 1 + (-8.32 + 3.26i)T + (22.7 - 21.0i)T^{2} \)
37 \( 1 + (-6.18 - 3.56i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.91 - 3.12i)T + (9.12 + 39.9i)T^{2} \)
47 \( 1 + (1.53 + 6.74i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (-0.152 + 2.03i)T + (-52.4 - 7.89i)T^{2} \)
59 \( 1 + (-2.69 + 1.29i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (6.59 + 2.58i)T + (44.7 + 41.4i)T^{2} \)
67 \( 1 + (-2.95 - 0.912i)T + (55.3 + 37.7i)T^{2} \)
71 \( 1 + (-4.67 - 5.04i)T + (-5.30 + 70.8i)T^{2} \)
73 \( 1 + (5.47 - 0.410i)T + (72.1 - 10.8i)T^{2} \)
79 \( 1 + (-8.56 + 4.94i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.14 + 0.172i)T + (79.3 - 24.4i)T^{2} \)
89 \( 1 + (0.00618 - 0.000931i)T + (85.0 - 26.2i)T^{2} \)
97 \( 1 + (-2.14 - 0.490i)T + (87.3 + 42.0i)T^{2} \)
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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

−10.01898290303421598425052760721, −9.560944570578864612927724875512, −8.235416129554315638440328945549, −7.938402826455121810652016178647, −7.17732579601612446531233978212, −6.12937467029510538139342882144, −6.00968370863943019791484996770, −4.55348550338731579099283792361, −2.12619268588626503064479737646, −0.924166628958743846664942549219, 0.73031665730179382982773247100, 2.72120135336873237019903205547, 3.25573551879093612360208693514, 4.53855939251591798545369775922, 5.30667462677981408014158949926, 7.33361957131876623859476884025, 8.018777961595370735246277559223, 9.075645646875514254577344236293, 9.636767759178380413731420973195, 10.36510009490998917594361166429

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