Properties

Degree 2
Conductor 43
Sign $1$
Motivic weight 5
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9.57·2-s − 7.84·3-s + 59.7·4-s + 28.1·5-s − 75.1·6-s + 195.·7-s + 265.·8-s − 181.·9-s + 269.·10-s + 72.8·11-s − 468.·12-s + 301.·13-s + 1.87e3·14-s − 220.·15-s + 632.·16-s − 1.20e3·17-s − 1.73e3·18-s − 2.35e3·19-s + 1.67e3·20-s − 1.53e3·21-s + 697.·22-s + 516.·23-s − 2.08e3·24-s − 2.33e3·25-s + 2.88e3·26-s + 3.32e3·27-s + 1.16e4·28-s + ⋯
L(s)  = 1  + 1.69·2-s − 0.503·3-s + 1.86·4-s + 0.502·5-s − 0.851·6-s + 1.50·7-s + 1.46·8-s − 0.746·9-s + 0.851·10-s + 0.181·11-s − 0.939·12-s + 0.495·13-s + 2.55·14-s − 0.252·15-s + 0.617·16-s − 1.01·17-s − 1.26·18-s − 1.49·19-s + 0.938·20-s − 0.759·21-s + 0.307·22-s + 0.203·23-s − 0.738·24-s − 0.747·25-s + 0.838·26-s + 0.878·27-s + 2.81·28-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

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  $\chi_{43} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5/2),\ 1)\)
\(L(3)\)  \(\approx\)  \(3.82996\)
\(L(\frac12)\)  \(\approx\)  \(3.82996\)
\(L(\frac{7}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 43$,\(F_p(T)\) is a polynomial of degree 2. If $p = 43$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad43 \( 1 - 1.84e3T \)
good2 \( 1 - 9.57T + 32T^{2} \)
3 \( 1 + 7.84T + 243T^{2} \)
5 \( 1 - 28.1T + 3.12e3T^{2} \)
7 \( 1 - 195.T + 1.68e4T^{2} \)
11 \( 1 - 72.8T + 1.61e5T^{2} \)
13 \( 1 - 301.T + 3.71e5T^{2} \)
17 \( 1 + 1.20e3T + 1.41e6T^{2} \)
19 \( 1 + 2.35e3T + 2.47e6T^{2} \)
23 \( 1 - 516.T + 6.43e6T^{2} \)
29 \( 1 - 1.53e3T + 2.05e7T^{2} \)
31 \( 1 - 1.12e3T + 2.86e7T^{2} \)
37 \( 1 + 9.33e3T + 6.93e7T^{2} \)
41 \( 1 - 1.97e4T + 1.15e8T^{2} \)
47 \( 1 + 1.37e4T + 2.29e8T^{2} \)
53 \( 1 + 3.35e3T + 4.18e8T^{2} \)
59 \( 1 - 2.51e3T + 7.14e8T^{2} \)
61 \( 1 - 4.92e4T + 8.44e8T^{2} \)
67 \( 1 - 9.11e3T + 1.35e9T^{2} \)
71 \( 1 - 4.33e4T + 1.80e9T^{2} \)
73 \( 1 - 8.00e4T + 2.07e9T^{2} \)
79 \( 1 + 6.59e4T + 3.07e9T^{2} \)
83 \( 1 + 7.68e4T + 3.93e9T^{2} \)
89 \( 1 + 7.57e4T + 5.58e9T^{2} \)
97 \( 1 + 6.79e4T + 8.58e9T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−14.57146851302731399038201568742, −13.92723585224097393704268649693, −12.73295649132939892728768907874, −11.46006607273858791318664702885, −10.93241658553871058751711759994, −8.491264535247900348021733673668, −6.51137752190500011814673145751, −5.43434478862698674201748772362, −4.30807811987455266889374395728, −2.15692329361905417967922491558, 2.15692329361905417967922491558, 4.30807811987455266889374395728, 5.43434478862698674201748772362, 6.51137752190500011814673145751, 8.491264535247900348021733673668, 10.93241658553871058751711759994, 11.46006607273858791318664702885, 12.73295649132939892728768907874, 13.92723585224097393704268649693, 14.57146851302731399038201568742

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