Properties

Degree 2
Conductor 43
Sign $1$
Motivic weight 3
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.29·2-s + 1.43·3-s − 6.31·4-s + 20.4·5-s + 1.85·6-s + 29.9·7-s − 18.5·8-s − 24.9·9-s + 26.5·10-s − 22.8·11-s − 9.02·12-s − 44.4·13-s + 38.9·14-s + 29.1·15-s + 26.3·16-s − 13.0·17-s − 32.4·18-s + 5.41·19-s − 128.·20-s + 42.8·21-s − 29.7·22-s − 175.·23-s − 26.6·24-s + 291.·25-s − 57.7·26-s − 74.3·27-s − 189.·28-s + ⋯
L(s)  = 1  + 0.459·2-s + 0.275·3-s − 0.788·4-s + 1.82·5-s + 0.126·6-s + 1.61·7-s − 0.822·8-s − 0.924·9-s + 0.838·10-s − 0.626·11-s − 0.217·12-s − 0.948·13-s + 0.743·14-s + 0.502·15-s + 0.411·16-s − 0.186·17-s − 0.424·18-s + 0.0653·19-s − 1.44·20-s + 0.445·21-s − 0.288·22-s − 1.58·23-s − 0.226·24-s + 2.33·25-s − 0.435·26-s − 0.529·27-s − 1.27·28-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  $\chi_{43} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3/2),\ 1)\)
\(L(2)\)  \(\approx\)  \(1.85720\)
\(L(\frac12)\)  \(\approx\)  \(1.85720\)
\(L(\frac{5}{2})\)   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 \( 1 + 43T \)
good2 \( 1 - 1.29T + 8T^{2} \)
3 \( 1 - 1.43T + 27T^{2} \)
5 \( 1 - 20.4T + 125T^{2} \)
7 \( 1 - 29.9T + 343T^{2} \)
11 \( 1 + 22.8T + 1.33e3T^{2} \)
13 \( 1 + 44.4T + 2.19e3T^{2} \)
17 \( 1 + 13.0T + 4.91e3T^{2} \)
19 \( 1 - 5.41T + 6.85e3T^{2} \)
23 \( 1 + 175.T + 1.21e4T^{2} \)
29 \( 1 - 165.T + 2.43e4T^{2} \)
31 \( 1 + 155.T + 2.97e4T^{2} \)
37 \( 1 + 95.3T + 5.06e4T^{2} \)
41 \( 1 - 189.T + 6.89e4T^{2} \)
47 \( 1 + 37.2T + 1.03e5T^{2} \)
53 \( 1 - 559.T + 1.48e5T^{2} \)
59 \( 1 + 82.3T + 2.05e5T^{2} \)
61 \( 1 + 640.T + 2.26e5T^{2} \)
67 \( 1 + 509.T + 3.00e5T^{2} \)
71 \( 1 - 792.T + 3.57e5T^{2} \)
73 \( 1 - 612.T + 3.89e5T^{2} \)
79 \( 1 - 237.T + 4.93e5T^{2} \)
83 \( 1 - 418.T + 5.71e5T^{2} \)
89 \( 1 - 113.T + 7.04e5T^{2} \)
97 \( 1 - 1.64e3T + 9.12e5T^{2} \)
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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

−14.79709794572835449568059610920, −14.15254406534191890544270250523, −13.60132840758735207196911310843, −12.16606504029831010349935300838, −10.49441856012556909916677896583, −9.280423986309143826902337390269, −8.150115608227947422150358821166, −5.76253988828029033383401738990, −4.92099093914120585181442148797, −2.28025810147586165925541206680, 2.28025810147586165925541206680, 4.92099093914120585181442148797, 5.76253988828029033383401738990, 8.150115608227947422150358821166, 9.280423986309143826902337390269, 10.49441856012556909916677896583, 12.16606504029831010349935300838, 13.60132840758735207196911310843, 14.15254406534191890544270250523, 14.79709794572835449568059610920

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