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Properties

Label 2-43-43.7-c4-0-1
Degree $2$
Conductor $43$
Sign $-0.198 + 0.980i$
Analytic cond. $4.44490$
Root an. cond. $2.10829$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.69i·2-s + (−13.9 + 8.07i)3-s − 16.4·4-s + (2.48 − 1.43i)5-s + (−46.0 − 79.6i)6-s + (31.0 + 17.9i)7-s − 2.41i·8-s + (90.0 − 156. i)9-s + (8.15 + 14.1i)10-s − 132.·11-s + (229. − 132. i)12-s + (−17.1 + 29.6i)13-s + (−102. + 176. i)14-s + (−23.1 + 40.0i)15-s − 249.·16-s + (182. − 315. i)17-s + ⋯
L(s)  = 1  + 1.42i·2-s + (−1.55 + 0.897i)3-s − 1.02·4-s + (0.0992 − 0.0572i)5-s + (−1.27 − 2.21i)6-s + (0.633 + 0.365i)7-s − 0.0376i·8-s + (1.11 − 1.92i)9-s + (0.0815 + 0.141i)10-s − 1.09·11-s + (1.59 − 0.921i)12-s + (−0.101 + 0.175i)13-s + (−0.520 + 0.901i)14-s + (−0.102 + 0.178i)15-s − 0.972·16-s + (0.630 − 1.09i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.198 + 0.980i$
Analytic conductor: \(4.44490\)
Root analytic conductor: \(2.10829\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :2),\ -0.198 + 0.980i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.325391 - 0.398021i\)
\(L(\frac12)\) \(\approx\) \(0.325391 - 0.398021i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-1.77e3 - 522. i)T \)
good2 \( 1 - 5.69iT - 16T^{2} \)
3 \( 1 + (13.9 - 8.07i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (-2.48 + 1.43i)T + (312.5 - 541. i)T^{2} \)
7 \( 1 + (-31.0 - 17.9i)T + (1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + 132.T + 1.46e4T^{2} \)
13 \( 1 + (17.1 - 29.6i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 + (-182. + 315. i)T + (-4.17e4 - 7.23e4i)T^{2} \)
19 \( 1 + (495. - 286. i)T + (6.51e4 - 1.12e5i)T^{2} \)
23 \( 1 + (119. + 207. i)T + (-1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (-1.20e3 - 696. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (-290. - 503. i)T + (-4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + (1.68e3 - 974. i)T + (9.37e5 - 1.62e6i)T^{2} \)
41 \( 1 + 619.T + 2.82e6T^{2} \)
47 \( 1 + 1.84e3T + 4.87e6T^{2} \)
53 \( 1 + (409. + 708. i)T + (-3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + 1.00e3T + 1.21e7T^{2} \)
61 \( 1 + (3.11e3 + 1.79e3i)T + (6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (-3.85e3 - 6.68e3i)T + (-1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (-2.08e3 - 1.20e3i)T + (1.27e7 + 2.20e7i)T^{2} \)
73 \( 1 + (-5.81e3 - 3.35e3i)T + (1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (6.19e3 - 1.07e4i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (-3.48e3 - 6.03e3i)T + (-2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 + (-2.79e3 + 1.61e3i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 + 526.T + 8.85e7T^{2} \)
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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

−16.03823316215334470530463168839, −15.39342776966938226250475512464, −14.24128272308769593181962518123, −12.37671896499714355913470222309, −11.18430313688996747915041137939, −10.03595433341650648561739593058, −8.348671871642887434163401155580, −6.73749970452441444950943075135, −5.48210638668599050330855463249, −4.81159402570520889703188872200, 0.39265994367886636902542305150, 1.99428696664366832439953696563, 4.65751277631423284685742268478, 6.27007714551552020948396541018, 7.898354663659372953158503155765, 10.39470990537776579885522487621, 10.77997035023348702161767222392, 11.95985826555618130736406853724, 12.69622192975385828677971162589, 13.63724667910632700141859121313

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