Properties

Label 2-43-1.1-c5-0-12
Degree $2$
Conductor $43$
Sign $-1$
Analytic cond. $6.89650$
Root an. cond. $2.62611$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.08·2-s + 25.6·3-s − 15.2·4-s − 61.4·5-s − 104.·6-s − 184.·7-s + 193.·8-s + 415.·9-s + 251.·10-s − 130.·11-s − 392.·12-s − 1.17e3·13-s + 755.·14-s − 1.57e3·15-s − 300.·16-s − 493.·17-s − 1.69e3·18-s + 2.42e3·19-s + 939.·20-s − 4.74e3·21-s + 532.·22-s − 4.18e3·23-s + 4.96e3·24-s + 648.·25-s + 4.81e3·26-s + 4.42e3·27-s + 2.82e3·28-s + ⋯
L(s)  = 1  − 0.722·2-s + 1.64·3-s − 0.477·4-s − 1.09·5-s − 1.18·6-s − 1.42·7-s + 1.06·8-s + 1.70·9-s + 0.793·10-s − 0.324·11-s − 0.786·12-s − 1.93·13-s + 1.02·14-s − 1.80·15-s − 0.293·16-s − 0.413·17-s − 1.23·18-s + 1.54·19-s + 0.525·20-s − 2.34·21-s + 0.234·22-s − 1.64·23-s + 1.75·24-s + 0.207·25-s + 1.39·26-s + 1.16·27-s + 0.681·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-1$
Analytic conductor: \(6.89650\)
Root analytic conductor: \(2.62611\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 43,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + 1.84e3T \)
good2 \( 1 + 4.08T + 32T^{2} \)
3 \( 1 - 25.6T + 243T^{2} \)
5 \( 1 + 61.4T + 3.12e3T^{2} \)
7 \( 1 + 184.T + 1.68e4T^{2} \)
11 \( 1 + 130.T + 1.61e5T^{2} \)
13 \( 1 + 1.17e3T + 3.71e5T^{2} \)
17 \( 1 + 493.T + 1.41e6T^{2} \)
19 \( 1 - 2.42e3T + 2.47e6T^{2} \)
23 \( 1 + 4.18e3T + 6.43e6T^{2} \)
29 \( 1 - 1.39e3T + 2.05e7T^{2} \)
31 \( 1 - 3.16e3T + 2.86e7T^{2} \)
37 \( 1 - 1.29e4T + 6.93e7T^{2} \)
41 \( 1 - 9.52e3T + 1.15e8T^{2} \)
47 \( 1 + 9.42e3T + 2.29e8T^{2} \)
53 \( 1 + 5.07e3T + 4.18e8T^{2} \)
59 \( 1 + 1.29e4T + 7.14e8T^{2} \)
61 \( 1 + 8.99e3T + 8.44e8T^{2} \)
67 \( 1 + 4.10e4T + 1.35e9T^{2} \)
71 \( 1 + 2.44e4T + 1.80e9T^{2} \)
73 \( 1 - 2.34e4T + 2.07e9T^{2} \)
79 \( 1 - 2.18e4T + 3.07e9T^{2} \)
83 \( 1 + 4.04e4T + 3.93e9T^{2} \)
89 \( 1 + 1.14e5T + 5.58e9T^{2} \)
97 \( 1 + 2.66e4T + 8.58e9T^{2} \)
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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

−14.36466466235201857242294397082, −13.35902060276869033839389810794, −12.21608385651620847506418961805, −9.863639491492754043962941697451, −9.500085875473751635818595778262, −8.025586570974153264247902662077, −7.41033298985865075771267265819, −4.23206128678616044934236371680, −2.84609349549639997940067384663, 0, 2.84609349549639997940067384663, 4.23206128678616044934236371680, 7.41033298985865075771267265819, 8.025586570974153264247902662077, 9.500085875473751635818595778262, 9.863639491492754043962941697451, 12.21608385651620847506418961805, 13.35902060276869033839389810794, 14.36466466235201857242294397082

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