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Properties

Label 2-43-43.7-c4-0-6
Degree $2$
Conductor $43$
Sign $0.505 - 0.862i$
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  + 2.98i·2-s + (−1.27 + 0.734i)3-s + 7.11·4-s + (26.8 − 15.5i)5-s + (−2.18 − 3.79i)6-s + (15.5 + 8.97i)7-s + 68.9i·8-s + (−39.4 + 68.2i)9-s + (46.2 + 80.0i)10-s + 38.7·11-s + (−9.04 + 5.22i)12-s + (29.5 − 51.1i)13-s + (−26.7 + 46.3i)14-s + (−22.7 + 39.4i)15-s − 91.5·16-s + (7.85 − 13.6i)17-s + ⋯
L(s)  = 1  + 0.745i·2-s + (−0.141 + 0.0815i)3-s + 0.444·4-s + (1.07 − 0.620i)5-s + (−0.0607 − 0.105i)6-s + (0.317 + 0.183i)7-s + 1.07i·8-s + (−0.486 + 0.842i)9-s + (0.462 + 0.800i)10-s + 0.320·11-s + (−0.0628 + 0.0362i)12-s + (0.174 − 0.302i)13-s + (−0.136 + 0.236i)14-s + (−0.101 + 0.175i)15-s − 0.357·16-s + (0.0271 − 0.0470i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 - 0.862i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.505 - 0.862i$
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: \((2,\ 43,\ (\ :2),\ 0.505 - 0.862i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.63595 + 0.937836i\)
\(L(\frac12)\) \(\approx\) \(1.63595 + 0.937836i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (1.50e3 + 1.06e3i)T \)
good2 \( 1 - 2.98iT - 16T^{2} \)
3 \( 1 + (1.27 - 0.734i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (-26.8 + 15.5i)T + (312.5 - 541. i)T^{2} \)
7 \( 1 + (-15.5 - 8.97i)T + (1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 - 38.7T + 1.46e4T^{2} \)
13 \( 1 + (-29.5 + 51.1i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 + (-7.85 + 13.6i)T + (-4.17e4 - 7.23e4i)T^{2} \)
19 \( 1 + (-143. + 82.6i)T + (6.51e4 - 1.12e5i)T^{2} \)
23 \( 1 + (212. + 368. i)T + (-1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (251. + 145. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (904. + 1.56e3i)T + (-4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + (1.16e3 - 673. i)T + (9.37e5 - 1.62e6i)T^{2} \)
41 \( 1 - 923.T + 2.82e6T^{2} \)
47 \( 1 - 1.84e3T + 4.87e6T^{2} \)
53 \( 1 + (-2.12e3 - 3.68e3i)T + (-3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + 3.00e3T + 1.21e7T^{2} \)
61 \( 1 + (2.15e3 + 1.24e3i)T + (6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (1.95e3 + 3.37e3i)T + (-1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (269. + 155. i)T + (1.27e7 + 2.20e7i)T^{2} \)
73 \( 1 + (-1.59e3 - 920. i)T + (1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (1.81e3 - 3.13e3i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (-2.24e3 - 3.88e3i)T + (-2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 + (-1.15e4 + 6.64e3i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 - 9.10e3T + 8.85e7T^{2} \)
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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

−15.50853878002040120919905062578, −14.30304624719762712009333150315, −13.35869889296098626067861700946, −11.79190452395576217770682143721, −10.57561828059015437927817589958, −9.009408721190658237896799105588, −7.75450457139916230226567994713, −6.06703894902949158655452014601, −5.17469183984821813046940053543, −2.09998856077697918869340386879, 1.64723488298574959809035560231, 3.35433354998635354994615545909, 5.91180643028456018380625425628, 7.03202624938666647138470825472, 9.220439378053433789633374133356, 10.34149777735843012374136496832, 11.36909175162032578162045904125, 12.40468761412332179300446575378, 13.84282512126441708077306979561, 14.77524229456816425667924746626

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