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Properties

Label 2-43-43.6-c3-0-5
Degree $2$
Conductor $43$
Sign $0.896 - 0.443i$
Analytic cond. $2.53708$
Root an. cond. $1.59282$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.92·2-s + (−2.49 + 4.32i)3-s + 16.2·4-s + (−1.06 + 1.83i)5-s + (−12.2 + 21.2i)6-s + (−14.5 − 25.1i)7-s + 40.6·8-s + (1.04 + 1.80i)9-s + (−5.22 + 9.05i)10-s + 23.0·11-s + (−40.5 + 70.2i)12-s + (−31.8 − 55.2i)13-s + (−71.5 − 123. i)14-s + (−5.29 − 9.17i)15-s + 70.3·16-s + (8.60 + 14.9i)17-s + ⋯
L(s)  = 1  + 1.74·2-s + (−0.480 + 0.831i)3-s + 2.03·4-s + (−0.0949 + 0.164i)5-s + (−0.836 + 1.44i)6-s + (−0.784 − 1.35i)7-s + 1.79·8-s + (0.0386 + 0.0669i)9-s + (−0.165 + 0.286i)10-s + 0.631·11-s + (−0.976 + 1.69i)12-s + (−0.680 − 1.17i)13-s + (−1.36 − 2.36i)14-s + (−0.0912 − 0.157i)15-s + 1.09·16-s + (0.122 + 0.212i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.443i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.896 - 0.443i$
Analytic conductor: \(2.53708\)
Root analytic conductor: \(1.59282\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :3/2),\ 0.896 - 0.443i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.54659 + 0.595461i\)
\(L(\frac12)\) \(\approx\) \(2.54659 + 0.595461i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-101. + 262. i)T \)
good2 \( 1 - 4.92T + 8T^{2} \)
3 \( 1 + (2.49 - 4.32i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (1.06 - 1.83i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (14.5 + 25.1i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 - 23.0T + 1.33e3T^{2} \)
13 \( 1 + (31.8 + 55.2i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-8.60 - 14.9i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (71.0 - 123. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-19.1 + 33.2i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-108. - 188. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-120. + 208. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (13.6 - 23.6i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 175.T + 6.89e4T^{2} \)
47 \( 1 - 25.7T + 1.03e5T^{2} \)
53 \( 1 + (-97.5 + 168. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + 781.T + 2.05e5T^{2} \)
61 \( 1 + (-347. - 602. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-14.5 + 25.1i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (470. + 815. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (-118. - 204. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (85.9 + 148. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (159. - 275. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-418. + 725. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 665.T + 9.12e5T^{2} \)
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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

−15.28808512071138825784117915196, −14.40825543137793791513472823579, −13.22771117738470445752703058678, −12.33722870467117531908828337793, −10.81495346132323407988581863081, −10.16080853896655121388574510949, −7.31875162636572253496321732994, −5.95819136802201148333744215210, −4.50705785427279156227302410701, −3.48657340899656343379139245994, 2.54318369666163534472028602598, 4.60483193290624109274371872325, 6.20791930432370365574835303888, 6.79135856678585399497146506635, 9.196235789564348287675416692357, 11.53516769563134665850507046669, 12.18991877215894425880759925365, 12.82034438642558408753556411828, 14.02560740980384452141660420553, 15.19994151247943683185685789310

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