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Properties

Label 2-43-43.36-c5-0-5
Degree $2$
Conductor $43$
Sign $-0.221 + 0.975i$
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  − 9.23·2-s + (−13.7 − 23.8i)3-s + 53.2·4-s + (45.8 + 79.3i)5-s + (127. + 219. i)6-s + (101. − 175. i)7-s − 196.·8-s + (−256. + 444. i)9-s + (−423. − 732. i)10-s + 347.·11-s + (−732. − 1.26e3i)12-s + (76.3 − 132. i)13-s + (−934. + 1.61e3i)14-s + (1.26e3 − 2.18e3i)15-s + 110.·16-s + (402. − 697. i)17-s + ⋯
L(s)  = 1  − 1.63·2-s + (−0.882 − 1.52i)3-s + 1.66·4-s + (0.819 + 1.41i)5-s + (1.44 + 2.49i)6-s + (0.780 − 1.35i)7-s − 1.08·8-s + (−1.05 + 1.83i)9-s + (−1.33 − 2.31i)10-s + 0.865·11-s + (−1.46 − 2.54i)12-s + (0.125 − 0.216i)13-s + (−1.27 + 2.20i)14-s + (1.44 − 2.50i)15-s + 0.108·16-s + (0.338 − 0.585i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.221 + 0.975i$
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: \((2,\ 43,\ (\ :5/2),\ -0.221 + 0.975i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.402663 - 0.504379i\)
\(L(\frac12)\) \(\approx\) \(0.402663 - 0.504379i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-9.97e3 + 6.89e3i)T \)
good2 \( 1 + 9.23T + 32T^{2} \)
3 \( 1 + (13.7 + 23.8i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (-45.8 - 79.3i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (-101. + 175. i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 - 347.T + 1.61e5T^{2} \)
13 \( 1 + (-76.3 + 132. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + (-402. + 697. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-243. - 422. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (423. + 734. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (-2.69e3 + 4.66e3i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 + (3.28e3 + 5.69e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-475. - 823. i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 1.74e3T + 1.15e8T^{2} \)
47 \( 1 + 1.70e4T + 2.29e8T^{2} \)
53 \( 1 + (4.57e3 + 7.92e3i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 - 2.50e4T + 7.14e8T^{2} \)
61 \( 1 + (-1.46e4 + 2.53e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (3.28e4 + 5.68e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (2.24e4 - 3.88e4i)T + (-9.02e8 - 1.56e9i)T^{2} \)
73 \( 1 + (2.15e4 - 3.72e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-2.90e4 + 5.03e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (-1.49e4 - 2.59e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 + (-2.54e4 - 4.41e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 6.42e4T + 8.58e9T^{2} \)
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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

−14.37290784675419876465116414348, −13.54492484513993651644849377735, −11.59494957971302125758991073496, −10.93082929479757576013200729389, −9.948773094952992476172273507144, −7.84844656042601446824025431586, −7.11832739145625974682196264322, −6.26360735417828268839590671269, −1.98038258088835788169132958138, −0.75803461063867115005919089324, 1.38921528151110977946421782548, 4.83431529005394036938503541012, 5.92893681454296379465891773335, 8.734547022539242732717701021964, 9.038686213526890371219409820956, 10.05870779771565825606935157692, 11.32018849161317539047333700629, 12.24179399747188655693516465977, 14.70562425814870846622453668514, 16.03321344496230799160706669709

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