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Properties

Label 2-43-43.6-c5-0-9
Degree $2$
Conductor $43$
Sign $0.803 + 0.595i$
Analytic cond. $6.89650$
Root an. cond. $2.62611$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 10.9·2-s + (8.65 − 14.9i)3-s + 88.5·4-s + (−5.02 + 8.70i)5-s + (−94.9 + 164. i)6-s + (36.4 + 63.0i)7-s − 620.·8-s + (−28.1 − 48.8i)9-s + (55.1 − 95.5i)10-s + 500.·11-s + (766. − 1.32e3i)12-s + (376. + 652. i)13-s + (−399. − 692. i)14-s + (86.9 + 150. i)15-s + 3.98e3·16-s + (−829. − 1.43e3i)17-s + ⋯
L(s)  = 1  − 1.94·2-s + (0.554 − 0.961i)3-s + 2.76·4-s + (−0.0898 + 0.155i)5-s + (−1.07 + 1.86i)6-s + (0.280 + 0.486i)7-s − 3.43·8-s + (−0.115 − 0.200i)9-s + (0.174 − 0.302i)10-s + 1.24·11-s + (1.53 − 2.66i)12-s + (0.618 + 1.07i)13-s + (−0.545 − 0.944i)14-s + (0.0997 + 0.172i)15-s + 3.89·16-s + (−0.695 − 1.20i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 + 0.595i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.803 + 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.803 + 0.595i$
Analytic conductor: \(6.89650\)
Root analytic conductor: \(2.62611\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :5/2),\ 0.803 + 0.595i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.867489 - 0.286633i\)
\(L(\frac12)\) \(\approx\) \(0.867489 - 0.286633i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (1.20e4 - 1.25e3i)T \)
good2 \( 1 + 10.9T + 32T^{2} \)
3 \( 1 + (-8.65 + 14.9i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (5.02 - 8.70i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (-36.4 - 63.0i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 - 500.T + 1.61e5T^{2} \)
13 \( 1 + (-376. - 652. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + (829. + 1.43e3i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-1.13e3 + 1.95e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-20.0 + 34.8i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (638. + 1.10e3i)T + (-1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (-2.82e3 + 4.89e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-1.52e3 + 2.64e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + 2.61e3T + 1.15e8T^{2} \)
47 \( 1 - 2.10e4T + 2.29e8T^{2} \)
53 \( 1 + (1.70e4 - 2.95e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 - 1.74e4T + 7.14e8T^{2} \)
61 \( 1 + (-448. - 776. i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (2.05e4 - 3.56e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (-1.09e4 - 1.89e4i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 + (1.94e4 + 3.36e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (-4.59e4 - 7.96e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-3.91e4 + 6.78e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + (2.84e4 - 4.92e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + 331.T + 8.58e9T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.17750585706487130123123718792, −13.72914202976911477091616720431, −11.84439193336793768672010254221, −11.24181473315337074836568093117, −9.324615670235054865878679831030, −8.776977199963228494325921279614, −7.35711691908663946452870455939, −6.63636127171587717584494508843, −2.46116495518237126842423396104, −1.14898757751196740382604175540, 1.23583604715974703179350674389, 3.55515684624322465241762083119, 6.45945477097910191460441110295, 8.110207217678430252812717638998, 8.874865911923126972907998977079, 10.05769210097158026944374835082, 10.74049031311049767200235509772, 12.16333898002687231811960342909, 14.52675142982842077560372890094, 15.47564053521055804642999870134

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