Properties

Degree 2
Conductor 43
Sign $0.253 + 0.967i$
Motivic weight 9
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.33·2-s + (−9.83 + 17.0i)3-s − 493.·4-s + (215. − 374. i)5-s + (−42.6 + 73.9i)6-s + (2.23e3 + 3.86e3i)7-s − 4.36e3·8-s + (9.64e3 + 1.67e4i)9-s + (936. − 1.62e3i)10-s − 2.74e4·11-s + (4.85e3 − 8.40e3i)12-s + (−4.60e4 − 7.96e4i)13-s + (9.67e3 + 1.67e4i)14-s + (4.24e3 + 7.35e3i)15-s + 2.33e5·16-s + (−2.00e5 − 3.47e5i)17-s + ⋯
L(s)  = 1  + 0.191·2-s + (−0.0701 + 0.121i)3-s − 0.963·4-s + (0.154 − 0.267i)5-s + (−0.0134 + 0.0232i)6-s + (0.351 + 0.608i)7-s − 0.376·8-s + (0.490 + 0.848i)9-s + (0.0296 − 0.0513i)10-s − 0.564·11-s + (0.0675 − 0.116i)12-s + (−0.446 − 0.773i)13-s + (0.0673 + 0.116i)14-s + (0.0216 + 0.0375i)15-s + 0.891·16-s + (−0.583 − 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.253 + 0.967i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.253 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.253 + 0.967i$
motivic weight  =  \(9\)
character  :  $\chi_{43} (6, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :9/2),\ 0.253 + 0.967i)\)
\(L(5)\)  \(\approx\)  \(0.966421 - 0.746021i\)
\(L(\frac12)\)  \(\approx\)  \(0.966421 - 0.746021i\)
\(L(\frac{11}{2})\)   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 \( 1 + (1.88e7 + 1.21e7i)T \)
good2 \( 1 - 4.33T + 512T^{2} \)
3 \( 1 + (9.83 - 17.0i)T + (-9.84e3 - 1.70e4i)T^{2} \)
5 \( 1 + (-215. + 374. i)T + (-9.76e5 - 1.69e6i)T^{2} \)
7 \( 1 + (-2.23e3 - 3.86e3i)T + (-2.01e7 + 3.49e7i)T^{2} \)
11 \( 1 + 2.74e4T + 2.35e9T^{2} \)
13 \( 1 + (4.60e4 + 7.96e4i)T + (-5.30e9 + 9.18e9i)T^{2} \)
17 \( 1 + (2.00e5 + 3.47e5i)T + (-5.92e10 + 1.02e11i)T^{2} \)
19 \( 1 + (-3.90e5 + 6.75e5i)T + (-1.61e11 - 2.79e11i)T^{2} \)
23 \( 1 + (-5.76e5 + 9.97e5i)T + (-9.00e11 - 1.55e12i)T^{2} \)
29 \( 1 + (1.36e6 + 2.36e6i)T + (-7.25e12 + 1.25e13i)T^{2} \)
31 \( 1 + (-2.80e6 + 4.85e6i)T + (-1.32e13 - 2.28e13i)T^{2} \)
37 \( 1 + (-5.55e6 + 9.61e6i)T + (-6.49e13 - 1.12e14i)T^{2} \)
41 \( 1 - 8.96e6T + 3.27e14T^{2} \)
47 \( 1 + 3.27e7T + 1.11e15T^{2} \)
53 \( 1 + (2.28e7 - 3.95e7i)T + (-1.64e15 - 2.85e15i)T^{2} \)
59 \( 1 - 9.98e7T + 8.66e15T^{2} \)
61 \( 1 + (5.24e7 + 9.08e7i)T + (-5.84e15 + 1.01e16i)T^{2} \)
67 \( 1 + (1.37e8 - 2.38e8i)T + (-1.36e16 - 2.35e16i)T^{2} \)
71 \( 1 + (-1.39e8 - 2.42e8i)T + (-2.29e16 + 3.97e16i)T^{2} \)
73 \( 1 + (-8.73e7 - 1.51e8i)T + (-2.94e16 + 5.09e16i)T^{2} \)
79 \( 1 + (1.14e8 + 1.97e8i)T + (-5.99e16 + 1.03e17i)T^{2} \)
83 \( 1 + (-3.61e8 + 6.26e8i)T + (-9.34e16 - 1.61e17i)T^{2} \)
89 \( 1 + (-4.13e8 + 7.16e8i)T + (-1.75e17 - 3.03e17i)T^{2} \)
97 \( 1 - 1.08e9T + 7.60e17T^{2} \)
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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

−13.46780147923348996518987880610, −12.91714265196086923798344972890, −11.37143140754048305466839173626, −9.932010345497387266223515233152, −8.846056287411006104141069372859, −7.55675794117131983373475256663, −5.35574812315613643687053704803, −4.70060426086382572939323317229, −2.60527728602465310796968382154, −0.46571649233901808448794144785, 1.27887611637413925467790020094, 3.58357475232398006669311803266, 4.82823165005702951975533604672, 6.48012437248320298614052403504, 7.982818162029670136729298448852, 9.413613145786792136156442979082, 10.45333115901992452839894805282, 12.08011619834282240564408652954, 13.12978849487364876279427312334, 14.18142589956284098280239458476

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