Properties

Degree 2
Conductor 43
Sign $-0.900 - 0.434i$
Motivic weight 3
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.05 + 4.61i)2-s + (0.0821 + 0.0762i)3-s + (−12.9 + 6.24i)4-s + (−9.16 + 1.38i)5-s + (−0.265 + 0.459i)6-s + (13.3 + 23.1i)7-s + (−18.8 − 23.6i)8-s + (−2.01 − 26.9i)9-s + (−16.0 − 40.8i)10-s + (46.6 + 22.4i)11-s + (−1.54 − 0.475i)12-s + (7.16 − 18.2i)13-s + (−92.6 + 85.9i)14-s + (−0.858 − 0.585i)15-s + (17.5 − 21.9i)16-s + (25.7 + 3.88i)17-s + ⋯
L(s)  = 1  + (0.372 + 1.63i)2-s + (0.0158 + 0.0146i)3-s + (−1.62 + 0.780i)4-s + (−0.819 + 0.123i)5-s + (−0.0180 + 0.0312i)6-s + (0.720 + 1.24i)7-s + (−0.834 − 1.04i)8-s + (−0.0746 − 0.996i)9-s + (−0.506 − 1.29i)10-s + (1.27 + 0.615i)11-s + (−0.0370 − 0.0114i)12-s + (0.152 − 0.389i)13-s + (−1.76 + 1.64i)14-s + (−0.0147 − 0.0100i)15-s + (0.273 − 0.343i)16-s + (0.367 + 0.0553i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.434i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.900 - 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.900 - 0.434i$
motivic weight  =  \(3\)
character  :  $\chi_{43} (10, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :3/2),\ -0.900 - 0.434i)\)
\(L(2)\)  \(\approx\)  \(0.312018 + 1.36402i\)
\(L(\frac12)\)  \(\approx\)  \(0.312018 + 1.36402i\)
\(L(\frac{5}{2})\)   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 \( 1 + (-91.6 + 266. i)T \)
good2 \( 1 + (-1.05 - 4.61i)T + (-7.20 + 3.47i)T^{2} \)
3 \( 1 + (-0.0821 - 0.0762i)T + (2.01 + 26.9i)T^{2} \)
5 \( 1 + (9.16 - 1.38i)T + (119. - 36.8i)T^{2} \)
7 \( 1 + (-13.3 - 23.1i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-46.6 - 22.4i)T + (829. + 1.04e3i)T^{2} \)
13 \( 1 + (-7.16 + 18.2i)T + (-1.61e3 - 1.49e3i)T^{2} \)
17 \( 1 + (-25.7 - 3.88i)T + (4.69e3 + 1.44e3i)T^{2} \)
19 \( 1 + (2.15 - 28.6i)T + (-6.78e3 - 1.02e3i)T^{2} \)
23 \( 1 + (-101. + 69.2i)T + (4.44e3 - 1.13e4i)T^{2} \)
29 \( 1 + (-152. + 141. i)T + (1.82e3 - 2.43e4i)T^{2} \)
31 \( 1 + (251. + 77.7i)T + (2.46e4 + 1.67e4i)T^{2} \)
37 \( 1 + (36.8 - 63.8i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-37.6 - 164. i)T + (-6.20e4 + 2.99e4i)T^{2} \)
47 \( 1 + (-447. + 215. i)T + (6.47e4 - 8.11e4i)T^{2} \)
53 \( 1 + (197. + 502. i)T + (-1.09e5 + 1.01e5i)T^{2} \)
59 \( 1 + (481. - 604. i)T + (-4.57e4 - 2.00e5i)T^{2} \)
61 \( 1 + (-150. + 46.4i)T + (1.87e5 - 1.27e5i)T^{2} \)
67 \( 1 + (-11.8 + 158. i)T + (-2.97e5 - 4.48e4i)T^{2} \)
71 \( 1 + (-796. - 543. i)T + (1.30e5 + 3.33e5i)T^{2} \)
73 \( 1 + (341. - 869. i)T + (-2.85e5 - 2.64e5i)T^{2} \)
79 \( 1 + (-130. - 225. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (463. + 430. i)T + (4.27e4 + 5.70e5i)T^{2} \)
89 \( 1 + (-275. - 255. i)T + (5.26e4 + 7.02e5i)T^{2} \)
97 \( 1 + (-891. - 429. i)T + (5.69e5 + 7.13e5i)T^{2} \)
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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

−15.50774865994047226816893091851, −15.03836742837622385510136403163, −14.34926782348257943531511523838, −12.50844021529217395626253378007, −11.64089192282719536590479214592, −9.182125838936056902539542114030, −8.238205885357219813784462264783, −6.90852726634691497560095904500, −5.65500072388307057775122382388, −4.06755542899030581705861978611, 1.24798980462878822709838682821, 3.62739965189245170599214120558, 4.70593019091015788659976322908, 7.47361027852601353743196283532, 9.073634830924813148843392419742, 10.81962367286215529529469071027, 11.16390852930208849461071762751, 12.34780050404110458424429107958, 13.75515649473441221863449870425, 14.26644822353308066030861864123

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