Properties

Degree 2
Conductor 43
Sign $-0.933 + 0.358i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.02 + 4.19i)2-s + (−0.581 − 0.0435i)3-s + (−3.56 − 4.46i)4-s + (−5.63 + 18.2i)5-s + (1.35 − 2.35i)6-s + (−16.3 + 9.46i)7-s + (−46.7 + 10.6i)8-s + (−79.7 − 12.0i)9-s + (−65.2 − 60.5i)10-s + (141. − 177. i)11-s + (1.87 + 2.75i)12-s + (−146. + 135. i)13-s + (−6.58 − 87.9i)14-s + (4.07 − 10.3i)15-s + (70.0 − 306. i)16-s + (−325. + 100. i)17-s + ⋯
L(s)  = 1  + (−0.505 + 1.04i)2-s + (−0.0646 − 0.00484i)3-s + (−0.222 − 0.279i)4-s + (−0.225 + 0.730i)5-s + (0.0377 − 0.0653i)6-s + (−0.334 + 0.193i)7-s + (−0.730 + 0.166i)8-s + (−0.984 − 0.148i)9-s + (−0.652 − 0.605i)10-s + (1.16 − 1.46i)11-s + (0.0130 + 0.0191i)12-s + (−0.867 + 0.804i)13-s + (−0.0336 − 0.448i)14-s + (0.0180 − 0.0461i)15-s + (0.273 − 1.19i)16-s + (−1.12 + 0.347i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 + 0.358i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.933 + 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.933 + 0.358i$
motivic weight  =  \(4\)
character  :  $\chi_{43} (3, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :2),\ -0.933 + 0.358i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.113262 - 0.611490i\)
\(L(\frac12)\)  \(\approx\)  \(0.113262 - 0.611490i\)
\(L(3)\)   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 \( 1 + (-1.09e3 - 1.49e3i)T \)
good2 \( 1 + (2.02 - 4.19i)T + (-9.97 - 12.5i)T^{2} \)
3 \( 1 + (0.581 + 0.0435i)T + (80.0 + 12.0i)T^{2} \)
5 \( 1 + (5.63 - 18.2i)T + (-516. - 352. i)T^{2} \)
7 \( 1 + (16.3 - 9.46i)T + (1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (-141. + 177. i)T + (-3.25e3 - 1.42e4i)T^{2} \)
13 \( 1 + (146. - 135. i)T + (2.13e3 - 2.84e4i)T^{2} \)
17 \( 1 + (325. - 100. i)T + (6.90e4 - 4.70e4i)T^{2} \)
19 \( 1 + (-97.3 - 646. i)T + (-1.24e5 + 3.84e4i)T^{2} \)
23 \( 1 + (-102. - 261. i)T + (-2.05e5 + 1.90e5i)T^{2} \)
29 \( 1 + (-1.20e3 + 90.2i)T + (6.99e5 - 1.05e5i)T^{2} \)
31 \( 1 + (899. - 613. i)T + (3.37e5 - 8.59e5i)T^{2} \)
37 \( 1 + (405. + 233. i)T + (9.37e5 + 1.62e6i)T^{2} \)
41 \( 1 + (-1.14e3 - 550. i)T + (1.76e6 + 2.20e6i)T^{2} \)
47 \( 1 + (1.83e3 + 2.30e3i)T + (-1.08e6 + 4.75e6i)T^{2} \)
53 \( 1 + (1.19e3 + 1.11e3i)T + (5.89e5 + 7.86e6i)T^{2} \)
59 \( 1 + (678. - 2.97e3i)T + (-1.09e7 - 5.25e6i)T^{2} \)
61 \( 1 + (-909. + 1.33e3i)T + (-5.05e6 - 1.28e7i)T^{2} \)
67 \( 1 + (-5.41e3 + 815. i)T + (1.92e7 - 5.93e6i)T^{2} \)
71 \( 1 + (4.55e3 + 1.78e3i)T + (1.86e7 + 1.72e7i)T^{2} \)
73 \( 1 + (-1.42e3 - 1.53e3i)T + (-2.12e6 + 2.83e7i)T^{2} \)
79 \( 1 + (3.53e3 + 6.13e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (402. - 5.36e3i)T + (-4.69e7 - 7.07e6i)T^{2} \)
89 \( 1 + (-4.27e3 - 320. i)T + (6.20e7 + 9.35e6i)T^{2} \)
97 \( 1 + (6.00e3 - 7.52e3i)T + (-1.96e7 - 8.63e7i)T^{2} \)
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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

−16.11624435556994833777514920884, −14.64125475140033213045550463620, −14.21886470038983369239701628196, −12.07955553204896072669697086694, −11.16613648405758398724347072569, −9.282595062373270041193672378634, −8.326828681410649922136416307103, −6.78600664553029154857581812775, −5.97206375460995582114520211043, −3.26396862764205523477651155983, 0.45699368658506254893983356788, 2.55652856001076577037322179406, 4.72029535963844521522472230941, 6.81614328201823799215954723007, 8.803617405626526389411145096194, 9.611009829589868981027214534986, 11.00019053578086699805238024037, 12.03931882000160183869389910894, 12.86762876989467792719413231500, 14.59528616618511598922245734440

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