Properties

Label 2-43-43.6-c9-0-7
Degree $2$
Conductor $43$
Sign $0.770 - 0.637i$
Analytic cond. $22.1465$
Root an. cond. $4.70601$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 30.5·2-s + (46.1 − 79.8i)3-s + 421.·4-s + (286. − 495. i)5-s + (−1.40e3 + 2.43e3i)6-s + (319. + 552. i)7-s + 2.77e3·8-s + (5.58e3 + 9.68e3i)9-s + (−8.74e3 + 1.51e4i)10-s − 5.06e4·11-s + (1.94e4 − 3.36e4i)12-s + (2.05e4 + 3.55e4i)13-s + (−9.74e3 − 1.68e4i)14-s + (−2.63e4 − 4.57e4i)15-s − 3.00e5·16-s + (−2.50e5 − 4.33e5i)17-s + ⋯
L(s)  = 1  − 1.34·2-s + (0.328 − 0.569i)3-s + 0.822·4-s + (0.204 − 0.354i)5-s + (−0.443 + 0.768i)6-s + (0.0502 + 0.0869i)7-s + 0.239·8-s + (0.283 + 0.491i)9-s + (−0.276 + 0.478i)10-s − 1.04·11-s + (0.270 − 0.468i)12-s + (0.199 + 0.345i)13-s + (−0.0678 − 0.117i)14-s + (−0.134 − 0.233i)15-s − 1.14·16-s + (−0.726 − 1.25i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.770 - 0.637i$
Analytic conductor: \(22.1465\)
Root analytic conductor: \(4.70601\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :9/2),\ 0.770 - 0.637i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.764364 + 0.275242i\)
\(L(\frac12)\) \(\approx\) \(0.764364 + 0.275242i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (3.06e6 - 2.22e7i)T \)
good2 \( 1 + 30.5T + 512T^{2} \)
3 \( 1 + (-46.1 + 79.8i)T + (-9.84e3 - 1.70e4i)T^{2} \)
5 \( 1 + (-286. + 495. i)T + (-9.76e5 - 1.69e6i)T^{2} \)
7 \( 1 + (-319. - 552. i)T + (-2.01e7 + 3.49e7i)T^{2} \)
11 \( 1 + 5.06e4T + 2.35e9T^{2} \)
13 \( 1 + (-2.05e4 - 3.55e4i)T + (-5.30e9 + 9.18e9i)T^{2} \)
17 \( 1 + (2.50e5 + 4.33e5i)T + (-5.92e10 + 1.02e11i)T^{2} \)
19 \( 1 + (5.44e3 - 9.42e3i)T + (-1.61e11 - 2.79e11i)T^{2} \)
23 \( 1 + (9.92e5 - 1.71e6i)T + (-9.00e11 - 1.55e12i)T^{2} \)
29 \( 1 + (-2.72e6 - 4.72e6i)T + (-7.25e12 + 1.25e13i)T^{2} \)
31 \( 1 + (-2.35e6 + 4.07e6i)T + (-1.32e13 - 2.28e13i)T^{2} \)
37 \( 1 + (6.16e6 - 1.06e7i)T + (-6.49e13 - 1.12e14i)T^{2} \)
41 \( 1 - 2.38e7T + 3.27e14T^{2} \)
47 \( 1 - 2.78e6T + 1.11e15T^{2} \)
53 \( 1 + (-1.50e7 + 2.60e7i)T + (-1.64e15 - 2.85e15i)T^{2} \)
59 \( 1 - 1.06e7T + 8.66e15T^{2} \)
61 \( 1 + (-6.27e7 - 1.08e8i)T + (-5.84e15 + 1.01e16i)T^{2} \)
67 \( 1 + (1.86e7 - 3.23e7i)T + (-1.36e16 - 2.35e16i)T^{2} \)
71 \( 1 + (6.97e7 + 1.20e8i)T + (-2.29e16 + 3.97e16i)T^{2} \)
73 \( 1 + (-1.12e8 - 1.94e8i)T + (-2.94e16 + 5.09e16i)T^{2} \)
79 \( 1 + (-2.61e8 - 4.53e8i)T + (-5.99e16 + 1.03e17i)T^{2} \)
83 \( 1 + (-1.64e8 + 2.85e8i)T + (-9.34e16 - 1.61e17i)T^{2} \)
89 \( 1 + (2.96e7 - 5.14e7i)T + (-1.75e17 - 3.03e17i)T^{2} \)
97 \( 1 - 7.07e8T + 7.60e17T^{2} \)
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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

−13.80771794294980390922375605525, −13.10358680612337152550398469486, −11.41425331335542567232159610571, −10.18296883780551044379818058527, −9.061438821639612326612868265813, −7.991594312136143384248447435514, −7.06659810466410574668722971333, −4.96185176646505111696196559268, −2.36730579351249679378335814492, −1.10526659717133451558909954327, 0.50410964058822940872769182821, 2.37027007228976751966560957152, 4.29736018983861246004208136636, 6.44882877016100654028841644517, 8.002130876354766235356442104995, 8.909968887860121425572672182823, 10.31477546604094468836261715756, 10.56808401599684221602996151212, 12.56919373933749565631384710008, 14.02835179847313484317985837347

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