Properties

Label 2-43-43.9-c9-0-29
Degree $2$
Conductor $43$
Sign $-0.948 - 0.316i$
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  + (5.76 + 7.22i)2-s + (−210. − 31.7i)3-s + (94.9 − 415. i)4-s + (−491. − 335. i)5-s + (−982. − 1.70e3i)6-s + (3.30e3 − 5.71e3i)7-s + (7.81e3 − 3.76e3i)8-s + (2.44e4 + 7.53e3i)9-s + (−410. − 5.48e3i)10-s + (−1.17e4 − 5.12e4i)11-s + (−3.31e4 + 8.44e4i)12-s + (−31.9 + 426. i)13-s + (6.03e4 − 9.09e3i)14-s + (9.27e4 + 8.60e4i)15-s + (−1.24e5 − 5.99e4i)16-s + (2.20e5 − 1.50e5i)17-s + ⋯
L(s)  = 1  + (0.254 + 0.319i)2-s + (−1.49 − 0.225i)3-s + (0.185 − 0.812i)4-s + (−0.351 − 0.239i)5-s + (−0.309 − 0.536i)6-s + (0.519 − 0.900i)7-s + (0.674 − 0.324i)8-s + (1.24 + 0.382i)9-s + (−0.0129 − 0.173i)10-s + (−0.241 − 1.05i)11-s + (−0.461 + 1.17i)12-s + (−0.000310 + 0.00413i)13-s + (0.419 − 0.0632i)14-s + (0.473 + 0.439i)15-s + (−0.475 − 0.228i)16-s + (0.639 − 0.435i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.948 - 0.316i$
Analytic conductor: \(22.1465\)
Root analytic conductor: \(4.70601\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :9/2),\ -0.948 - 0.316i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.0896630 + 0.551717i\)
\(L(\frac12)\) \(\approx\) \(0.0896630 + 0.551717i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (5.61e6 - 2.17e7i)T \)
good2 \( 1 + (-5.76 - 7.22i)T + (-113. + 499. i)T^{2} \)
3 \( 1 + (210. + 31.7i)T + (1.88e4 + 5.80e3i)T^{2} \)
5 \( 1 + (491. + 335. i)T + (7.13e5 + 1.81e6i)T^{2} \)
7 \( 1 + (-3.30e3 + 5.71e3i)T + (-2.01e7 - 3.49e7i)T^{2} \)
11 \( 1 + (1.17e4 + 5.12e4i)T + (-2.12e9 + 1.02e9i)T^{2} \)
13 \( 1 + (31.9 - 426. i)T + (-1.04e10 - 1.58e9i)T^{2} \)
17 \( 1 + (-2.20e5 + 1.50e5i)T + (4.33e10 - 1.10e11i)T^{2} \)
19 \( 1 + (6.20e5 - 1.91e5i)T + (2.66e11 - 1.81e11i)T^{2} \)
23 \( 1 + (1.13e6 - 1.05e6i)T + (1.34e11 - 1.79e12i)T^{2} \)
29 \( 1 + (-7.24e5 + 1.09e5i)T + (1.38e13 - 4.27e12i)T^{2} \)
31 \( 1 + (5.03e4 - 1.28e5i)T + (-1.93e13 - 1.79e13i)T^{2} \)
37 \( 1 + (-4.10e6 - 7.11e6i)T + (-6.49e13 + 1.12e14i)T^{2} \)
41 \( 1 + (8.51e6 + 1.06e7i)T + (-7.28e13 + 3.19e14i)T^{2} \)
47 \( 1 + (9.61e6 - 4.21e7i)T + (-1.00e15 - 4.85e14i)T^{2} \)
53 \( 1 + (2.62e6 + 3.50e7i)T + (-3.26e15 + 4.91e14i)T^{2} \)
59 \( 1 + (-1.02e8 - 4.94e7i)T + (5.40e15 + 6.77e15i)T^{2} \)
61 \( 1 + (-3.35e7 - 8.54e7i)T + (-8.57e15 + 7.95e15i)T^{2} \)
67 \( 1 + (-1.52e8 + 4.70e7i)T + (2.24e16 - 1.53e16i)T^{2} \)
71 \( 1 + (1.94e8 + 1.80e8i)T + (3.42e15 + 4.57e16i)T^{2} \)
73 \( 1 + (-1.70e7 + 2.27e8i)T + (-5.82e16 - 8.77e15i)T^{2} \)
79 \( 1 + (3.67e7 - 6.37e7i)T + (-5.99e16 - 1.03e17i)T^{2} \)
83 \( 1 + (5.07e8 + 7.65e7i)T + (1.78e17 + 5.51e16i)T^{2} \)
89 \( 1 + (-6.94e8 - 1.04e8i)T + (3.34e17 + 1.03e17i)T^{2} \)
97 \( 1 + (-4.04e7 - 1.77e8i)T + (-6.84e17 + 3.29e17i)T^{2} \)
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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

−13.38681841960017870546294581604, −11.88333231536714611397371323950, −10.99877300352801496111311531192, −10.17070545665214684126244023922, −7.88295558842782750621052876667, −6.49180511505714843743606911939, −5.54210339440717412772958042303, −4.36131038712166156613639235528, −1.23699768127443324659948289721, −0.23798362313260301446598304766, 2.12533898304796366293478792556, 4.15145958305455580168840509713, 5.34280457378301596219135835018, 6.85076244178084642561962031639, 8.287506955037826168053819155646, 10.28179712844288364128172559691, 11.36445168389796097682075968912, 12.08883116131591790411033461808, 12.82269381325233380820304007879, 14.84528227341195230084613492290

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