Properties

Label 2-43-43.6-c5-0-4
Degree $2$
Conductor $43$
Sign $-0.710 - 0.703i$
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  + 6.70·2-s + (−7.23 + 12.5i)3-s + 12.9·4-s + (−40.0 + 69.3i)5-s + (−48.5 + 84.0i)6-s + (−74.8 − 129. i)7-s − 127.·8-s + (16.7 + 28.9i)9-s + (−268. + 464. i)10-s + 126.·11-s + (−93.4 + 161. i)12-s + (420. + 729. i)13-s + (−501. − 868. i)14-s + (−579. − 1.00e3i)15-s − 1.27e3·16-s + (1.12e3 + 1.94e3i)17-s + ⋯
L(s)  = 1  + 1.18·2-s + (−0.464 + 0.804i)3-s + 0.403·4-s + (−0.716 + 1.24i)5-s + (−0.550 + 0.952i)6-s + (−0.577 − 0.999i)7-s − 0.706·8-s + (0.0687 + 0.119i)9-s + (−0.848 + 1.46i)10-s + 0.315·11-s + (−0.187 + 0.324i)12-s + (0.690 + 1.19i)13-s + (−0.683 − 1.18i)14-s + (−0.665 − 1.15i)15-s − 1.24·16-s + (0.943 + 1.63i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.710 - 0.703i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.710 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.710 - 0.703i$
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: \((2,\ 43,\ (\ :5/2),\ -0.710 - 0.703i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.611983 + 1.48783i\)
\(L(\frac12)\) \(\approx\) \(0.611983 + 1.48783i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-1.21e4 - 462. i)T \)
good2 \( 1 - 6.70T + 32T^{2} \)
3 \( 1 + (7.23 - 12.5i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (40.0 - 69.3i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (74.8 + 129. i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 - 126.T + 1.61e5T^{2} \)
13 \( 1 + (-420. - 729. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + (-1.12e3 - 1.94e3i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-1.46e3 + 2.53e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (1.99e3 - 3.45e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (1.13e3 + 1.97e3i)T + (-1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (-1.14e3 + 1.97e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-1.25e3 + 2.17e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 - 192.T + 1.15e8T^{2} \)
47 \( 1 + 6.42e3T + 2.29e8T^{2} \)
53 \( 1 + (1.13e4 - 1.96e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + 1.73e4T + 7.14e8T^{2} \)
61 \( 1 + (-1.50e4 - 2.60e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-726. + 1.25e3i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (-3.45e4 - 5.98e4i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 + (6.36e3 + 1.10e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (-1.32e4 - 2.28e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-1.49e4 + 2.59e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + (-3.15e4 + 5.47e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 1.31e5T + 8.58e9T^{2} \)
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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

−15.30845271443475104041004699910, −14.17587623833993062264377422521, −13.34583360143106514332999467149, −11.66841622017078789070909683213, −10.92178431947797825648624173650, −9.646833326878343745494025034077, −7.30367353525533599940121144295, −6.03513220868438510420992082140, −4.18171557696806928263806147037, −3.54925716928676528640512571810, 0.65961911552688731003604556412, 3.44082392520672163417693293881, 5.19340929804151088228086255329, 6.12745966490650756690995149637, 8.022094653343095520115687458766, 9.409587871436116561318335525618, 11.96190333731773131015965470997, 12.24162417629952061688725924704, 12.93634716921204120182622207230, 14.27996770606356458563522133591

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