Properties

Label 2-43-43.36-c5-0-9
Degree $2$
Conductor $43$
Sign $0.424 + 0.905i$
Analytic cond. $6.89650$
Root an. cond. $2.62611$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.24·2-s + (−0.494 − 0.856i)3-s − 30.4·4-s + (24.3 + 42.1i)5-s + (0.616 + 1.06i)6-s + (83.1 − 144. i)7-s + 77.8·8-s + (121. − 209. i)9-s + (−30.3 − 52.6i)10-s − 436.·11-s + (15.0 + 26.0i)12-s + (486. − 842. i)13-s + (−103. + 179. i)14-s + (24.0 − 41.7i)15-s + 877.·16-s + (375. − 650. i)17-s + ⋯
L(s)  = 1  − 0.220·2-s + (−0.0317 − 0.0549i)3-s − 0.951·4-s + (0.435 + 0.754i)5-s + (0.00699 + 0.0121i)6-s + (0.641 − 1.11i)7-s + 0.430·8-s + (0.497 − 0.862i)9-s + (−0.0960 − 0.166i)10-s − 1.08·11-s + (0.0301 + 0.0522i)12-s + (0.798 − 1.38i)13-s + (−0.141 + 0.244i)14-s + (0.0276 − 0.0478i)15-s + 0.856·16-s + (0.315 − 0.545i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.424 + 0.905i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.424 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.424 + 0.905i$
Analytic conductor: \(6.89650\)
Root analytic conductor: \(2.62611\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :5/2),\ 0.424 + 0.905i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.01800 - 0.647321i\)
\(L(\frac12)\) \(\approx\) \(1.01800 - 0.647321i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-3.63e3 + 1.15e4i)T \)
good2 \( 1 + 1.24T + 32T^{2} \)
3 \( 1 + (0.494 + 0.856i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (-24.3 - 42.1i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (-83.1 + 144. i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + 436.T + 1.61e5T^{2} \)
13 \( 1 + (-486. + 842. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + (-375. + 650. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-283. - 490. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (498. + 863. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (1.99e3 - 3.46e3i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 + (-3.89e3 - 6.74e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (6.02e3 + 1.04e4i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 1.55e4T + 1.15e8T^{2} \)
47 \( 1 - 8.22e3T + 2.29e8T^{2} \)
53 \( 1 + (2.10e3 + 3.64e3i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 - 3.81e4T + 7.14e8T^{2} \)
61 \( 1 + (-1.40e4 + 2.44e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-5.91e3 - 1.02e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (3.77e4 - 6.53e4i)T + (-9.02e8 - 1.56e9i)T^{2} \)
73 \( 1 + (-2.78e4 + 4.81e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (3.89e4 - 6.74e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (-8.25e3 - 1.42e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 + (-3.81e4 - 6.61e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 1.06e3T + 8.58e9T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.49441191729158725608025306200, −13.67946705562467693775654654544, −12.63484614075823317561934900998, −10.58826703443503091558975895229, −10.16579632316849369501480135814, −8.408680100190340756780128416866, −7.17891100792717986070903455468, −5.30631301198236070707421642259, −3.56076196792263369072471550436, −0.800522247411870592612088235989, 1.73718995283540156506861307878, 4.58179402718384055270834952944, 5.56647025468892648988853561573, 8.025138311301018270954565804360, 8.886076081479075971360066429288, 10.05043793368702255477515560948, 11.61962641926300978387271134179, 13.10560430895601687360357447111, 13.67330804933204566164653181902, 15.21940670016295327547614478349

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