Properties

Label 2-43-43.10-c3-0-2
Degree $2$
Conductor $43$
Sign $-0.357 - 0.933i$
Analytic cond. $2.53708$
Root an. cond. $1.59282$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.882 + 3.86i)2-s + (4.09 + 3.79i)3-s + (−6.96 + 3.35i)4-s + (7.73 − 1.16i)5-s + (−11.0 + 19.1i)6-s + (−13.6 − 23.6i)7-s + (0.667 + 0.836i)8-s + (0.313 + 4.17i)9-s + (11.3 + 28.8i)10-s + (−32.0 − 15.4i)11-s + (−41.2 − 12.7i)12-s + (−4.10 + 10.4i)13-s + (79.3 − 73.5i)14-s + (36.0 + 24.5i)15-s + (−41.2 + 51.6i)16-s + (97.9 + 14.7i)17-s + ⋯
L(s)  = 1  + (0.312 + 1.36i)2-s + (0.787 + 0.731i)3-s + (−0.870 + 0.419i)4-s + (0.691 − 0.104i)5-s + (−0.753 + 1.30i)6-s + (−0.736 − 1.27i)7-s + (0.0294 + 0.0369i)8-s + (0.0116 + 0.154i)9-s + (0.358 + 0.912i)10-s + (−0.878 − 0.423i)11-s + (−0.992 − 0.306i)12-s + (−0.0876 + 0.223i)13-s + (1.51 − 1.40i)14-s + (0.620 + 0.423i)15-s + (−0.643 + 0.807i)16-s + (1.39 + 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.357 - 0.933i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.357 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.357 - 0.933i$
Analytic conductor: \(2.53708\)
Root analytic conductor: \(1.59282\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :3/2),\ -0.357 - 0.933i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.08018 + 1.57056i\)
\(L(\frac12)\) \(\approx\) \(1.08018 + 1.57056i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-249. + 131. i)T \)
good2 \( 1 + (-0.882 - 3.86i)T + (-7.20 + 3.47i)T^{2} \)
3 \( 1 + (-4.09 - 3.79i)T + (2.01 + 26.9i)T^{2} \)
5 \( 1 + (-7.73 + 1.16i)T + (119. - 36.8i)T^{2} \)
7 \( 1 + (13.6 + 23.6i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (32.0 + 15.4i)T + (829. + 1.04e3i)T^{2} \)
13 \( 1 + (4.10 - 10.4i)T + (-1.61e3 - 1.49e3i)T^{2} \)
17 \( 1 + (-97.9 - 14.7i)T + (4.69e3 + 1.44e3i)T^{2} \)
19 \( 1 + (-0.481 + 6.42i)T + (-6.78e3 - 1.02e3i)T^{2} \)
23 \( 1 + (-51.0 + 34.8i)T + (4.44e3 - 1.13e4i)T^{2} \)
29 \( 1 + (186. - 173. i)T + (1.82e3 - 2.43e4i)T^{2} \)
31 \( 1 + (-38.7 - 11.9i)T + (2.46e4 + 1.67e4i)T^{2} \)
37 \( 1 + (147. - 255. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (47.9 + 209. i)T + (-6.20e4 + 2.99e4i)T^{2} \)
47 \( 1 + (308. - 148. i)T + (6.47e4 - 8.11e4i)T^{2} \)
53 \( 1 + (132. + 337. i)T + (-1.09e5 + 1.01e5i)T^{2} \)
59 \( 1 + (-229. + 287. i)T + (-4.57e4 - 2.00e5i)T^{2} \)
61 \( 1 + (-750. + 231. i)T + (1.87e5 - 1.27e5i)T^{2} \)
67 \( 1 + (-27.9 + 372. i)T + (-2.97e5 - 4.48e4i)T^{2} \)
71 \( 1 + (537. + 366. i)T + (1.30e5 + 3.33e5i)T^{2} \)
73 \( 1 + (385. - 981. i)T + (-2.85e5 - 2.64e5i)T^{2} \)
79 \( 1 + (-431. - 747. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-389. - 361. i)T + (4.27e4 + 5.70e5i)T^{2} \)
89 \( 1 + (-793. - 736. i)T + (5.26e4 + 7.02e5i)T^{2} \)
97 \( 1 + (487. + 234. i)T + (5.69e5 + 7.13e5i)T^{2} \)
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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

−15.85717577518342764429598530265, −14.65741581684750264651185885534, −13.92840103352446207943899911032, −13.04698571900475060324304942740, −10.54445279159540700965107773073, −9.547399669692846802397564283511, −8.112529918162159072504284717557, −6.82374550803035430124345744552, −5.33324695774196559329775381531, −3.59916009190537388528736855325, 2.07589585139863515901110272956, 3.00473020310611555921518358548, 5.58791474746059120878393231371, 7.61689940023678363593611009506, 9.299616555310810439497545054281, 10.21183756024858223170720597121, 11.86108970363494943664435849872, 12.85615883607116161936258788754, 13.38313893367819279155763470663, 14.68783524657529618336258026073

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