Properties

Label 2-43-43.36-c5-0-2
Degree $2$
Conductor $43$
Sign $0.915 - 0.403i$
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  − 2.35·2-s + (−4.70 − 8.15i)3-s − 26.4·4-s + (15.5 + 26.9i)5-s + (11.1 + 19.2i)6-s + (−36.1 + 62.5i)7-s + 137.·8-s + (77.1 − 133. i)9-s + (−36.6 − 63.5i)10-s + 785.·11-s + (124. + 215. i)12-s + (−374. + 647. i)13-s + (85.1 − 147. i)14-s + (146. − 253. i)15-s + 521.·16-s + (198. − 343. i)17-s + ⋯
L(s)  = 1  − 0.416·2-s + (−0.302 − 0.523i)3-s − 0.826·4-s + (0.278 + 0.481i)5-s + (0.125 + 0.218i)6-s + (−0.278 + 0.482i)7-s + 0.761·8-s + (0.317 − 0.549i)9-s + (−0.115 − 0.200i)10-s + 1.95·11-s + (0.249 + 0.432i)12-s + (−0.613 + 1.06i)13-s + (0.116 − 0.201i)14-s + (0.168 − 0.291i)15-s + 0.508·16-s + (0.166 − 0.288i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 - 0.403i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.915 - 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.915 - 0.403i$
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: \((2,\ 43,\ (\ :5/2),\ 0.915 - 0.403i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.03035 + 0.216864i\)
\(L(\frac12)\) \(\approx\) \(1.03035 + 0.216864i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (1.14e4 + 3.89e3i)T \)
good2 \( 1 + 2.35T + 32T^{2} \)
3 \( 1 + (4.70 + 8.15i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (-15.5 - 26.9i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (36.1 - 62.5i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 - 785.T + 1.61e5T^{2} \)
13 \( 1 + (374. - 647. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + (-198. + 343. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-1.24e3 - 2.16e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-628. - 1.08e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (2.01e3 - 3.48e3i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 + (3.39e3 + 5.88e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-5.80e3 - 1.00e4i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 1.81e4T + 1.15e8T^{2} \)
47 \( 1 - 1.73e4T + 2.29e8T^{2} \)
53 \( 1 + (-1.00e4 - 1.74e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + 2.70e4T + 7.14e8T^{2} \)
61 \( 1 + (-2.36e4 + 4.10e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-1.85e4 - 3.21e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (2.51e4 - 4.36e4i)T + (-9.02e8 - 1.56e9i)T^{2} \)
73 \( 1 + (-1.76e4 + 3.05e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-4.90e3 + 8.49e3i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (2.43e4 + 4.22e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 + (3.67e4 + 6.37e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 1.46e4T + 8.58e9T^{2} \)
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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

−14.67563430823627052811955689307, −14.02976361303369831732399660686, −12.50799659127622999877780739156, −11.65304565635603031374367941621, −9.710943105141090623988974324129, −9.160295618765270538022851889571, −7.27290763352540513444591909746, −6.06480903016824738090527939237, −3.97978321739471823859328785207, −1.31382520435190640995557505380, 0.885911701304746528947442517914, 4.05269051570246505687004252197, 5.28614854814454756373908051029, 7.33291697385824232822177257090, 9.018547422721207755186688970487, 9.755702807683845356435205315386, 10.97940194865687329056811495806, 12.65370988545484472182849193826, 13.65035812171810483866126532008, 14.84714152328148185713693517487

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