Properties

Label 2-43-43.12-c8-0-20
Degree $2$
Conductor $43$
Sign $0.532 + 0.846i$
Analytic cond. $17.5172$
Root an. cond. $4.18536$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.99 + 8.29i)2-s + (7.77 + 11.4i)3-s + (106. − 133. i)4-s + (319. − 344. i)5-s + (−63.5 + 110. i)6-s + (−1.51e3 + 876. i)7-s + (3.83e3 + 875. i)8-s + (2.32e3 − 5.93e3i)9-s + (4.12e3 + 1.27e3i)10-s + (−1.42e4 − 1.79e4i)11-s + (2.35e3 + 176. i)12-s + (−3.68e4 + 1.13e4i)13-s + (−1.33e4 − 9.09e3i)14-s + (6.40e3 + 965. i)15-s + (−1.69e3 − 7.42e3i)16-s + (8.97e4 − 8.33e4i)17-s + ⋯
L(s)  = 1  + (0.249 + 0.518i)2-s + (0.0960 + 0.140i)3-s + (0.417 − 0.522i)4-s + (0.510 − 0.550i)5-s + (−0.0490 + 0.0849i)6-s + (−0.632 + 0.365i)7-s + (0.936 + 0.213i)8-s + (0.354 − 0.903i)9-s + (0.412 + 0.127i)10-s + (−0.976 − 1.22i)11-s + (0.113 + 0.00852i)12-s + (−1.28 + 0.397i)13-s + (−0.347 − 0.236i)14-s + (0.126 + 0.0190i)15-s + (−0.0258 − 0.113i)16-s + (1.07 − 0.997i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.532 + 0.846i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.532 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.532 + 0.846i$
Analytic conductor: \(17.5172\)
Root analytic conductor: \(4.18536\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :4),\ 0.532 + 0.846i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.99106 - 1.10034i\)
\(L(\frac12)\) \(\approx\) \(1.99106 - 1.10034i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-2.25e6 - 2.56e6i)T \)
good2 \( 1 + (-3.99 - 8.29i)T + (-159. + 200. i)T^{2} \)
3 \( 1 + (-7.77 - 11.4i)T + (-2.39e3 + 6.10e3i)T^{2} \)
5 \( 1 + (-319. + 344. i)T + (-2.91e4 - 3.89e5i)T^{2} \)
7 \( 1 + (1.51e3 - 876. i)T + (2.88e6 - 4.99e6i)T^{2} \)
11 \( 1 + (1.42e4 + 1.79e4i)T + (-4.76e7 + 2.08e8i)T^{2} \)
13 \( 1 + (3.68e4 - 1.13e4i)T + (6.73e8 - 4.59e8i)T^{2} \)
17 \( 1 + (-8.97e4 + 8.33e4i)T + (5.21e8 - 6.95e9i)T^{2} \)
19 \( 1 + (-1.35e4 + 5.32e3i)T + (1.24e10 - 1.15e10i)T^{2} \)
23 \( 1 + (-2.15e5 + 3.25e4i)T + (7.48e10 - 2.30e10i)T^{2} \)
29 \( 1 + (-3.35e5 + 4.91e5i)T + (-1.82e11 - 4.65e11i)T^{2} \)
31 \( 1 + (-5.39e4 + 7.19e5i)T + (-8.43e11 - 1.27e11i)T^{2} \)
37 \( 1 + (6.41e5 + 3.70e5i)T + (1.75e12 + 3.04e12i)T^{2} \)
41 \( 1 + (1.47e6 - 7.12e5i)T + (4.97e12 - 6.24e12i)T^{2} \)
47 \( 1 + (-2.48e6 + 3.11e6i)T + (-5.29e12 - 2.32e13i)T^{2} \)
53 \( 1 + (3.03e6 + 9.34e5i)T + (5.14e13 + 3.50e13i)T^{2} \)
59 \( 1 + (-3.86e6 - 1.69e7i)T + (-1.32e14 + 6.37e13i)T^{2} \)
61 \( 1 + (6.07e6 - 4.55e5i)T + (1.89e14 - 2.85e13i)T^{2} \)
67 \( 1 + (2.10e6 + 5.35e6i)T + (-2.97e14 + 2.76e14i)T^{2} \)
71 \( 1 + (6.36e6 - 4.22e7i)T + (-6.17e14 - 1.90e14i)T^{2} \)
73 \( 1 + (1.23e7 + 3.99e7i)T + (-6.66e14 + 4.54e14i)T^{2} \)
79 \( 1 + (-2.55e7 - 4.42e7i)T + (-7.58e14 + 1.31e15i)T^{2} \)
83 \( 1 + (-1.80e7 + 1.22e7i)T + (8.22e14 - 2.09e15i)T^{2} \)
89 \( 1 + (3.89e7 + 5.71e7i)T + (-1.43e15 + 3.66e15i)T^{2} \)
97 \( 1 + (-9.55e7 - 1.19e8i)T + (-1.74e15 + 7.64e15i)T^{2} \)
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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

−14.14607886582390624994040863632, −13.04257647091204985980649165087, −11.74302322438041285398272786605, −10.12028672311368542897137717362, −9.245445627869950041493055378506, −7.42772381801566005307747456378, −6.04199805332769850407163340477, −5.05051133137695192554870632216, −2.79407367880918888281893719439, −0.76929830284917593996952430039, 1.96635756994253533407143175741, 3.05139311873321312618549508956, 4.91578575654452797363594639547, 6.98407863481936824498815191972, 7.76987517264972925886740573574, 10.19770838389153248715646620911, 10.43381101653703947336292268705, 12.40670749479490395424812123422, 12.88644737575056155703641509782, 14.17709300018015170628939812300

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