Properties

Label 2-43-43.6-c3-0-6
Degree $2$
Conductor $43$
Sign $-0.757 + 0.652i$
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  − 4.23·2-s + (4.42 − 7.67i)3-s + 9.93·4-s + (−2.98 + 5.16i)5-s + (−18.7 + 32.4i)6-s + (−11.2 − 19.4i)7-s − 8.18·8-s + (−25.7 − 44.5i)9-s + (12.6 − 21.8i)10-s − 46.0·11-s + (43.9 − 76.2i)12-s + (−4.54 − 7.86i)13-s + (47.6 + 82.5i)14-s + (26.4 + 45.7i)15-s − 44.8·16-s + (40.1 + 69.4i)17-s + ⋯
L(s)  = 1  − 1.49·2-s + (0.852 − 1.47i)3-s + 1.24·4-s + (−0.266 + 0.462i)5-s + (−1.27 + 2.21i)6-s + (−0.607 − 1.05i)7-s − 0.361·8-s + (−0.953 − 1.65i)9-s + (0.399 − 0.691i)10-s − 1.26·11-s + (1.05 − 1.83i)12-s + (−0.0969 − 0.167i)13-s + (0.909 + 1.57i)14-s + (0.454 + 0.787i)15-s − 0.700·16-s + (0.572 + 0.991i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.757 + 0.652i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.757 + 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.757 + 0.652i$
Analytic conductor: \(2.53708\)
Root analytic conductor: \(1.59282\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :3/2),\ -0.757 + 0.652i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.214418 - 0.577836i\)
\(L(\frac12)\) \(\approx\) \(0.214418 - 0.577836i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (33.1 - 280. i)T \)
good2 \( 1 + 4.23T + 8T^{2} \)
3 \( 1 + (-4.42 + 7.67i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (2.98 - 5.16i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (11.2 + 19.4i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + 46.0T + 1.33e3T^{2} \)
13 \( 1 + (4.54 + 7.86i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-40.1 - 69.4i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-72.4 + 125. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-55.6 + 96.3i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-40.6 - 70.4i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-122. + 212. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-88.0 + 152. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 186.T + 6.89e4T^{2} \)
47 \( 1 - 29.1T + 1.03e5T^{2} \)
53 \( 1 + (-273. + 474. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + 281.T + 2.05e5T^{2} \)
61 \( 1 + (-349. - 605. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (15.3 - 26.5i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (172. + 298. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (-324. - 562. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-62.2 - 107. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (376. - 651. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-22.1 + 38.4i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.13e3T + 9.12e5T^{2} \)
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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

−15.10560035909600223073799137283, −13.59878248897867262851641997388, −12.90003376353069020020029379971, −11.08327819261426248828283580106, −9.924221172474153871177447065970, −8.453410404129622780306263067933, −7.50967256892038827621730171656, −6.86897267600754058636325689971, −2.78446486349931591025292863579, −0.73287063535844170945306195713, 2.93298052356304248668845125700, 5.13651350224421955417648170129, 7.85723566210354236749153129072, 8.765734707922334069564930846823, 9.685262434825404350052613698582, 10.34652176211689547832021671866, 11.99166741189397649839472071890, 13.88635780527911892688969833053, 15.51006498586159046101519433963, 15.91586487882735209990250405904

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