Properties

Label 2-43-43.36-c7-0-23
Degree $2$
Conductor $43$
Sign $-0.318 - 0.947i$
Analytic cond. $13.4325$
Root an. cond. $3.66504$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.06·2-s + (−34.1 − 59.1i)3-s − 126.·4-s + (−137. − 238. i)5-s + (−36.4 − 63.0i)6-s + (809. − 1.40e3i)7-s − 271.·8-s + (−1.23e3 + 2.13e3i)9-s + (−147. − 254. i)10-s − 5.26e3·11-s + (4.32e3 + 7.49e3i)12-s + (−2.05e3 + 3.55e3i)13-s + (863. − 1.49e3i)14-s + (−9.40e3 + 1.62e4i)15-s + 1.59e4·16-s + (1.09e4 − 1.89e4i)17-s + ⋯
L(s)  = 1  + 0.0943·2-s + (−0.729 − 1.26i)3-s − 0.991·4-s + (−0.492 − 0.853i)5-s + (−0.0688 − 0.119i)6-s + (0.892 − 1.54i)7-s − 0.187·8-s + (−0.564 + 0.978i)9-s + (−0.0464 − 0.0805i)10-s − 1.19·11-s + (0.723 + 1.25i)12-s + (−0.259 + 0.448i)13-s + (0.0841 − 0.145i)14-s + (−0.719 + 1.24i)15-s + 0.973·16-s + (0.540 − 0.935i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.318 - 0.947i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.318 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.318 - 0.947i$
Analytic conductor: \(13.4325\)
Root analytic conductor: \(3.66504\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :7/2),\ -0.318 - 0.947i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.301321 + 0.419212i\)
\(L(\frac12)\) \(\approx\) \(0.301321 + 0.419212i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-2.11e5 + 4.76e5i)T \)
good2 \( 1 - 1.06T + 128T^{2} \)
3 \( 1 + (34.1 + 59.1i)T + (-1.09e3 + 1.89e3i)T^{2} \)
5 \( 1 + (137. + 238. i)T + (-3.90e4 + 6.76e4i)T^{2} \)
7 \( 1 + (-809. + 1.40e3i)T + (-4.11e5 - 7.13e5i)T^{2} \)
11 \( 1 + 5.26e3T + 1.94e7T^{2} \)
13 \( 1 + (2.05e3 - 3.55e3i)T + (-3.13e7 - 5.43e7i)T^{2} \)
17 \( 1 + (-1.09e4 + 1.89e4i)T + (-2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (-2.46e4 - 4.26e4i)T + (-4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 + (9.04e3 + 1.56e4i)T + (-1.70e9 + 2.94e9i)T^{2} \)
29 \( 1 + (-4.02e4 + 6.97e4i)T + (-8.62e9 - 1.49e10i)T^{2} \)
31 \( 1 + (1.03e5 + 1.79e5i)T + (-1.37e10 + 2.38e10i)T^{2} \)
37 \( 1 + (-1.06e5 - 1.84e5i)T + (-4.74e10 + 8.22e10i)T^{2} \)
41 \( 1 - 3.20e5T + 1.94e11T^{2} \)
47 \( 1 + 1.30e6T + 5.06e11T^{2} \)
53 \( 1 + (-8.52e5 - 1.47e6i)T + (-5.87e11 + 1.01e12i)T^{2} \)
59 \( 1 + 8.57e5T + 2.48e12T^{2} \)
61 \( 1 + (-2.25e5 + 3.90e5i)T + (-1.57e12 - 2.72e12i)T^{2} \)
67 \( 1 + (-1.15e6 - 1.99e6i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 + (-4.77e5 + 8.26e5i)T + (-4.54e12 - 7.87e12i)T^{2} \)
73 \( 1 + (1.86e6 - 3.22e6i)T + (-5.52e12 - 9.56e12i)T^{2} \)
79 \( 1 + (9.99e5 - 1.73e6i)T + (-9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 + (6.66e5 + 1.15e6i)T + (-1.35e13 + 2.35e13i)T^{2} \)
89 \( 1 + (3.76e6 + 6.52e6i)T + (-2.21e13 + 3.83e13i)T^{2} \)
97 \( 1 - 9.17e5T + 8.07e13T^{2} \)
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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

−13.48574006457694709121213867056, −12.56561183786268311729593153964, −11.56527190549863684986511560465, −10.03328546229204423440222959359, −8.028648590460812505945760225005, −7.52294860207048713534920903531, −5.43029946812777532183890804985, −4.30268591946076485520741058602, −1.14940555216925373688842100905, −0.28655342878361600362474136994, 3.14168071344045170828174242309, 4.91341313663894929221083229082, 5.50389921341966054914974485460, 8.004653143543806117542480973356, 9.301517806322052860613064766771, 10.55406602365857392549727981059, 11.44080225539656250679591217270, 12.75421196045193405329144026053, 14.61389830070730533856146334706, 15.17988797178022730186233236209

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