Properties

Label 2-43-1.1-c1-0-0
Degree $2$
Conductor $43$
Sign $1$
Analytic cond. $0.343356$
Root an. cond. $0.585966$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·2-s + 1.41·3-s + 3.41·5-s − 2.00·6-s − 3.41·7-s + 2.82·8-s − 0.999·9-s − 4.82·10-s − 3.82·11-s − 1.82·13-s + 4.82·14-s + 4.82·15-s − 4.00·16-s + 2.17·17-s + 1.41·18-s + 0.828·19-s − 4.82·21-s + 5.41·22-s + 6.65·23-s + 4·24-s + 6.65·25-s + 2.58·26-s − 5.65·27-s − 4.24·29-s − 6.82·30-s + ⋯
L(s)  = 1  − 1.00·2-s + 0.816·3-s + 1.52·5-s − 0.816·6-s − 1.29·7-s + 0.999·8-s − 0.333·9-s − 1.52·10-s − 1.15·11-s − 0.507·13-s + 1.29·14-s + 1.24·15-s − 1.00·16-s + 0.526·17-s + 0.333·18-s + 0.190·19-s − 1.05·21-s + 1.15·22-s + 1.38·23-s + 0.816·24-s + 1.33·25-s + 0.507·26-s − 1.08·27-s − 0.787·29-s − 1.24·30-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $1$
Analytic conductor: \(0.343356\)
Root analytic conductor: \(0.585966\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6205398574\)
\(L(\frac12)\) \(\approx\) \(0.6205398574\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 - T \)
good2 \( 1 + 1.41T + 2T^{2} \)
3 \( 1 - 1.41T + 3T^{2} \)
5 \( 1 - 3.41T + 5T^{2} \)
7 \( 1 + 3.41T + 7T^{2} \)
11 \( 1 + 3.82T + 11T^{2} \)
13 \( 1 + 1.82T + 13T^{2} \)
17 \( 1 - 2.17T + 17T^{2} \)
19 \( 1 - 0.828T + 19T^{2} \)
23 \( 1 - 6.65T + 23T^{2} \)
29 \( 1 + 4.24T + 29T^{2} \)
31 \( 1 + 3T + 31T^{2} \)
37 \( 1 - 8.48T + 37T^{2} \)
41 \( 1 - 1.82T + 41T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 - 13.8T + 53T^{2} \)
59 \( 1 + 4.82T + 59T^{2} \)
61 \( 1 + 0.242T + 61T^{2} \)
67 \( 1 + 7.48T + 67T^{2} \)
71 \( 1 + 3.17T + 71T^{2} \)
73 \( 1 + 16.2T + 73T^{2} \)
79 \( 1 - 4.82T + 79T^{2} \)
83 \( 1 - 3.34T + 83T^{2} \)
89 \( 1 + 1.75T + 89T^{2} \)
97 \( 1 - 1.82T + 97T^{2} \)
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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

−16.41857875277764751285340800172, −14.76986745196960011746858392114, −13.49001219312746589168226351132, −13.05457630487850294800727128587, −10.52307576982547646640094397647, −9.608244660344817536413119381754, −9.010560501914790387819456190647, −7.44784426687994133111116572758, −5.61345574477677785453996129942, −2.68049063912669140610122847079, 2.68049063912669140610122847079, 5.61345574477677785453996129942, 7.44784426687994133111116572758, 9.010560501914790387819456190647, 9.608244660344817536413119381754, 10.52307576982547646640094397647, 13.05457630487850294800727128587, 13.49001219312746589168226351132, 14.76986745196960011746858392114, 16.41857875277764751285340800172

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