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Properties

Label 2-43-43.37-c4-0-9
Degree $2$
Conductor $43$
Sign $0.425 + 0.904i$
Analytic cond. $4.44490$
Root an. cond. $2.10829$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7.27i·2-s + (−9.67 − 5.58i)3-s − 36.8·4-s + (5.33 + 3.07i)5-s + (40.6 − 70.3i)6-s + (35.8 − 20.7i)7-s − 151. i·8-s + (21.9 + 38.0i)9-s + (−22.3 + 38.7i)10-s − 189.·11-s + (356. + 205. i)12-s + (−67.9 − 117. i)13-s + (150. + 260. i)14-s + (−34.4 − 59.5i)15-s + 512.·16-s + (−212. − 368. i)17-s + ⋯
L(s)  = 1  + 1.81i·2-s + (−1.07 − 0.620i)3-s − 2.30·4-s + (0.213 + 0.123i)5-s + (1.12 − 1.95i)6-s + (0.732 − 0.422i)7-s − 2.36i·8-s + (0.271 + 0.469i)9-s + (−0.223 + 0.387i)10-s − 1.56·11-s + (2.47 + 1.43i)12-s + (−0.401 − 0.696i)13-s + (0.768 + 1.33i)14-s + (−0.152 − 0.264i)15-s + 2.00·16-s + (−0.735 − 1.27i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.425 + 0.904i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.425 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.425 + 0.904i$
Analytic conductor: \(4.44490\)
Root analytic conductor: \(2.10829\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :2),\ 0.425 + 0.904i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.168400 - 0.106847i\)
\(L(\frac12)\) \(\approx\) \(0.168400 - 0.106847i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (1.59e3 - 928. i)T \)
good2 \( 1 - 7.27iT - 16T^{2} \)
3 \( 1 + (9.67 + 5.58i)T + (40.5 + 70.1i)T^{2} \)
5 \( 1 + (-5.33 - 3.07i)T + (312.5 + 541. i)T^{2} \)
7 \( 1 + (-35.8 + 20.7i)T + (1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + 189.T + 1.46e4T^{2} \)
13 \( 1 + (67.9 + 117. i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (212. + 368. i)T + (-4.17e4 + 7.23e4i)T^{2} \)
19 \( 1 + (-274. - 158. i)T + (6.51e4 + 1.12e5i)T^{2} \)
23 \( 1 + (465. - 805. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (696. - 402. i)T + (3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (-481. + 833. i)T + (-4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + (-546. - 315. i)T + (9.37e5 + 1.62e6i)T^{2} \)
41 \( 1 + 1.09e3T + 2.82e6T^{2} \)
47 \( 1 - 1.74e3T + 4.87e6T^{2} \)
53 \( 1 + (-1.53e3 + 2.66e3i)T + (-3.94e6 - 6.83e6i)T^{2} \)
59 \( 1 + 1.44e3T + 1.21e7T^{2} \)
61 \( 1 + (-1.82e3 + 1.05e3i)T + (6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (2.29e3 - 3.97e3i)T + (-1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + (-3.01e3 + 1.73e3i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (-2.82e3 + 1.62e3i)T + (1.41e7 - 2.45e7i)T^{2} \)
79 \( 1 + (5.34e3 + 9.26e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (4.49e3 - 7.79e3i)T + (-2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 + (9.66e3 + 5.58e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + 4.90e3T + 8.85e7T^{2} \)
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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

−15.30174161672044328406488585542, −13.91373818713992669135775869306, −13.12969260571943412454029079401, −11.52675060711240094604429596409, −9.892408528998960563148859509135, −7.936626492343341817737828803162, −7.27252453158889842479717913050, −5.78872219846856614814554664011, −5.02672718187988469348375738287, −0.14256182002331504236164515918, 2.19082666937320727167272687088, 4.43303231039668539979732592467, 5.43017334943629279278152099229, 8.473885022965827706127997724053, 9.980487745074767107781107169158, 10.77254438962748636061472315155, 11.58245742814091442194917011202, 12.59246404484891034268295991794, 13.76545080559638823856721153127, 15.40932073360424257227085614164

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