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Properties

Label 2-43-43.36-c5-0-10
Degree $2$
Conductor $43$
Sign $0.868 - 0.496i$
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  + 9.70·2-s + (7.50 + 13.0i)3-s + 62.1·4-s + (−9.53 − 16.5i)5-s + (72.8 + 126. i)6-s + (−13.4 + 23.2i)7-s + 292.·8-s + (8.78 − 15.2i)9-s + (−92.5 − 160. i)10-s − 218.·11-s + (466. + 807. i)12-s + (−136. + 236. i)13-s + (−130. + 226. i)14-s + (143. − 248. i)15-s + 846.·16-s + (386. − 669. i)17-s + ⋯
L(s)  = 1  + 1.71·2-s + (0.481 + 0.834i)3-s + 1.94·4-s + (−0.170 − 0.295i)5-s + (0.825 + 1.43i)6-s + (−0.103 + 0.179i)7-s + 1.61·8-s + (0.0361 − 0.0625i)9-s + (−0.292 − 0.506i)10-s − 0.544·11-s + (0.934 + 1.61i)12-s + (−0.224 + 0.388i)13-s + (−0.177 + 0.308i)14-s + (0.164 − 0.284i)15-s + 0.826·16-s + (0.324 − 0.561i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 - 0.496i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.868 - 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.868 - 0.496i$
Analytic conductor: \(6.89650\)
Root analytic conductor: \(2.62611\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :5/2),\ 0.868 - 0.496i)\)

Particular Values

\(L(3)\) \(\approx\) \(4.19195 + 1.11429i\)
\(L(\frac12)\) \(\approx\) \(4.19195 + 1.11429i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (1.18e4 + 2.67e3i)T \)
good2 \( 1 - 9.70T + 32T^{2} \)
3 \( 1 + (-7.50 - 13.0i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (9.53 + 16.5i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (13.4 - 23.2i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + 218.T + 1.61e5T^{2} \)
13 \( 1 + (136. - 236. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + (-386. + 669. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (605. + 1.04e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (422. + 731. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (3.60e3 - 6.25e3i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 + (100. + 174. i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-1.83e3 - 3.17e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 1.46e4T + 1.15e8T^{2} \)
47 \( 1 - 2.34e4T + 2.29e8T^{2} \)
53 \( 1 + (-1.55e4 - 2.69e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 - 3.45e4T + 7.14e8T^{2} \)
61 \( 1 + (-2.22e3 + 3.85e3i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-1.85e4 - 3.21e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (-4.55e3 + 7.89e3i)T + (-9.02e8 - 1.56e9i)T^{2} \)
73 \( 1 + (1.57e4 - 2.72e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-1.75e4 + 3.04e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (1.66e4 + 2.87e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 + (-2.14e3 - 3.71e3i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 4.93e3T + 8.58e9T^{2} \)
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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

−14.94017395966674969000818999089, −14.00449842249541549705137102447, −12.83788999932497759617281707648, −11.87872152885602629501362783222, −10.45522944285448703186475610098, −8.902254237432220342391453515631, −6.91357761676522329704319173088, −5.22771987133899910448880855266, −4.16304060497007387278821752962, −2.80707176736526932558503744342, 2.19260273122713745702606866656, 3.68241741346403695395627353473, 5.43372130694472159594527144908, 6.87792127306681729279709603150, 7.978336700128122469133172644978, 10.42771193323730362330194650163, 11.82245677438456207647371369979, 12.94109696641255049083498266332, 13.46741569984956399999870942314, 14.61240681531212016499058612838

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