Properties

Label 2-43-43.6-c5-0-12
Degree $2$
Conductor $43$
Sign $-0.838 + 0.545i$
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  − 6.28·2-s + (10.0 − 17.4i)3-s + 7.51·4-s + (38.9 − 67.5i)5-s + (−63.3 + 109. i)6-s + (4.83 + 8.37i)7-s + 153.·8-s + (−81.3 − 140. i)9-s + (−245. + 424. i)10-s − 186.·11-s + (75.7 − 131. i)12-s + (−333. − 577. i)13-s + (−30.3 − 52.6i)14-s + (−785. − 1.36e3i)15-s − 1.20e3·16-s + (−506. − 877. i)17-s + ⋯
L(s)  = 1  − 1.11·2-s + (0.646 − 1.11i)3-s + 0.234·4-s + (0.697 − 1.20i)5-s + (−0.718 + 1.24i)6-s + (0.0372 + 0.0645i)7-s + 0.850·8-s + (−0.334 − 0.580i)9-s + (−0.774 + 1.34i)10-s − 0.465·11-s + (0.151 − 0.262i)12-s + (−0.547 − 0.947i)13-s + (−0.0414 − 0.0717i)14-s + (−0.901 − 1.56i)15-s − 1.17·16-s + (−0.425 − 0.736i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.838 + 0.545i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.838 + 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.838 + 0.545i$
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: \((2,\ 43,\ (\ :5/2),\ -0.838 + 0.545i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.278451 - 0.937997i\)
\(L(\frac12)\) \(\approx\) \(0.278451 - 0.937997i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-3.01e3 + 1.17e4i)T \)
good2 \( 1 + 6.28T + 32T^{2} \)
3 \( 1 + (-10.0 + 17.4i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (-38.9 + 67.5i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (-4.83 - 8.37i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + 186.T + 1.61e5T^{2} \)
13 \( 1 + (333. + 577. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + (506. + 877. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (510. - 884. i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (359. - 622. i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (-2.47e3 - 4.28e3i)T + (-1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (1.88e3 - 3.26e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-7.59e3 + 1.31e4i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + 1.16e4T + 1.15e8T^{2} \)
47 \( 1 - 1.04e4T + 2.29e8T^{2} \)
53 \( 1 + (1.14e4 - 1.98e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 - 3.72e4T + 7.14e8T^{2} \)
61 \( 1 + (5.92e3 + 1.02e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-2.97e4 + 5.15e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (-4.85e3 - 8.40e3i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 + (-3.85e4 - 6.67e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (4.50e4 + 7.80e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-8.00e3 + 1.38e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + (-3.79e4 + 6.57e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + 8.09e4T + 8.58e9T^{2} \)
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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

−14.13191299320619247585716300443, −13.18199797067718281324787729700, −12.47429438872053852799907018627, −10.39697219554440223198364209893, −9.144369245422849399063655200499, −8.321951459896005875554453711257, −7.29375401041344177639779135644, −5.18314970343256085115200788671, −2.07054845272405804724290691882, −0.71956114971248323703577410171, 2.43932726295227813972819567919, 4.38288161436444154866526030522, 6.70827021804433836726103958305, 8.307404779016906227977160350228, 9.564471772542704868534064367399, 10.12887899586026789693249843908, 11.10517931943045402249731633081, 13.48571680837252761901230589980, 14.49731686610205667212805948015, 15.39807083794007853730844607192

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