Properties

Label 2-43-43.6-c5-0-1
Degree $2$
Conductor $43$
Sign $-0.886 + 0.462i$
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  + 3.69·2-s + (−11.8 + 20.4i)3-s − 18.3·4-s + (21.7 − 37.7i)5-s + (−43.6 + 75.6i)6-s + (−19.0 − 32.9i)7-s − 186.·8-s + (−158. − 273. i)9-s + (80.4 − 139. i)10-s − 444.·11-s + (216. − 375. i)12-s + (−340. − 588. i)13-s + (−70.3 − 121. i)14-s + (514. + 891. i)15-s − 99.9·16-s + (541. + 937. i)17-s + ⋯
L(s)  = 1  + 0.653·2-s + (−0.758 + 1.31i)3-s − 0.573·4-s + (0.389 − 0.674i)5-s + (−0.495 + 0.857i)6-s + (−0.146 − 0.254i)7-s − 1.02·8-s + (−0.650 − 1.12i)9-s + (0.254 − 0.440i)10-s − 1.10·11-s + (0.434 − 0.753i)12-s + (−0.558 − 0.966i)13-s + (−0.0959 − 0.166i)14-s + (0.590 + 1.02i)15-s − 0.0976·16-s + (0.454 + 0.786i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.886 + 0.462i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.886 + 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.886 + 0.462i$
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: \((2,\ 43,\ (\ :5/2),\ -0.886 + 0.462i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0364151 - 0.148617i\)
\(L(\frac12)\) \(\approx\) \(0.0364151 - 0.148617i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-4.13e3 + 1.13e4i)T \)
good2 \( 1 - 3.69T + 32T^{2} \)
3 \( 1 + (11.8 - 20.4i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (-21.7 + 37.7i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (19.0 + 32.9i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + 444.T + 1.61e5T^{2} \)
13 \( 1 + (340. + 588. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + (-541. - 937. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (1.27e3 - 2.20e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-498. + 862. i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (2.08e3 + 3.60e3i)T + (-1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (3.78e3 - 6.54e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (2.34e3 - 4.06e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + 6.61e3T + 1.15e8T^{2} \)
47 \( 1 + 7.39e3T + 2.29e8T^{2} \)
53 \( 1 + (-3.27e3 + 5.67e3i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + 1.87e3T + 7.14e8T^{2} \)
61 \( 1 + (1.93e4 + 3.34e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (1.59e4 - 2.76e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (-3.91e4 - 6.77e4i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 + (-2.84e4 - 4.93e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (1.98e4 + 3.44e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-2.04e4 + 3.54e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + (-1.44e3 + 2.50e3i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + 9.66e4T + 8.58e9T^{2} \)
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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

−15.51306045479766648747764441719, −14.63139626336246367234940769556, −13.10566163464860462595079320502, −12.36509861925256578185348444793, −10.53458393808417452213773666103, −9.882595243415778130963849162949, −8.399061359396433416449170998322, −5.67193428963493953096772009463, −5.05709325313784078198735488618, −3.69615495198956500821584130799, 0.07111515780253443552142099649, 2.52821136190692787932533063911, 5.03795153206826911744887144885, 6.26962313695697811506299953598, 7.41156690780182897498994093019, 9.295299666058713454024085975610, 11.03203344239481294526234371113, 12.23389575941453250812801033977, 13.11404978751871416956215145306, 13.89543266645139044194276594783

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