Properties

Label 2-43-43.4-c3-0-1
Degree $2$
Conductor $43$
Sign $-0.738 - 0.674i$
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  + (−2.32 + 2.90i)2-s + (3.61 + 4.53i)3-s + (−1.30 − 5.70i)4-s + (13.3 + 6.42i)5-s − 21.5·6-s − 12.2·7-s + (−7.21 − 3.47i)8-s + (−1.46 + 6.42i)9-s + (−49.6 + 23.9i)10-s + (−0.456 + 2.00i)11-s + (21.1 − 26.5i)12-s + (−8.58 − 4.13i)13-s + (28.3 − 35.5i)14-s + (19.1 + 83.7i)15-s + (68.9 − 33.2i)16-s + (66.7 − 32.1i)17-s + ⋯
L(s)  = 1  + (−0.820 + 1.02i)2-s + (0.695 + 0.872i)3-s + (−0.162 − 0.712i)4-s + (1.19 + 0.574i)5-s − 1.46·6-s − 0.659·7-s + (−0.318 − 0.153i)8-s + (−0.0543 + 0.238i)9-s + (−1.57 + 0.756i)10-s + (−0.0125 + 0.0548i)11-s + (0.508 − 0.637i)12-s + (−0.183 − 0.0882i)13-s + (0.540 − 0.678i)14-s + (0.328 + 1.44i)15-s + (1.07 − 0.519i)16-s + (0.952 − 0.458i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.738 - 0.674i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.738 - 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.738 - 0.674i$
Analytic conductor: \(2.53708\)
Root analytic conductor: \(1.59282\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :3/2),\ -0.738 - 0.674i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.422421 + 1.08941i\)
\(L(\frac12)\) \(\approx\) \(0.422421 + 1.08941i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (177. + 219. i)T \)
good2 \( 1 + (2.32 - 2.90i)T + (-1.78 - 7.79i)T^{2} \)
3 \( 1 + (-3.61 - 4.53i)T + (-6.00 + 26.3i)T^{2} \)
5 \( 1 + (-13.3 - 6.42i)T + (77.9 + 97.7i)T^{2} \)
7 \( 1 + 12.2T + 343T^{2} \)
11 \( 1 + (0.456 - 2.00i)T + (-1.19e3 - 577. i)T^{2} \)
13 \( 1 + (8.58 + 4.13i)T + (1.36e3 + 1.71e3i)T^{2} \)
17 \( 1 + (-66.7 + 32.1i)T + (3.06e3 - 3.84e3i)T^{2} \)
19 \( 1 + (-22.9 - 100. i)T + (-6.17e3 + 2.97e3i)T^{2} \)
23 \( 1 + (-32.7 + 143. i)T + (-1.09e4 - 5.27e3i)T^{2} \)
29 \( 1 + (63.7 - 79.9i)T + (-5.42e3 - 2.37e4i)T^{2} \)
31 \( 1 + (192. - 240. i)T + (-6.62e3 - 2.90e4i)T^{2} \)
37 \( 1 - 434.T + 5.06e4T^{2} \)
41 \( 1 + (-114. + 143. i)T + (-1.53e4 - 6.71e4i)T^{2} \)
47 \( 1 + (77.5 + 339. i)T + (-9.35e4 + 4.50e4i)T^{2} \)
53 \( 1 + (-180. + 86.6i)T + (9.28e4 - 1.16e5i)T^{2} \)
59 \( 1 + (-141. + 68.0i)T + (1.28e5 - 1.60e5i)T^{2} \)
61 \( 1 + (162. + 204. i)T + (-5.05e4 + 2.21e5i)T^{2} \)
67 \( 1 + (-9.64 - 42.2i)T + (-2.70e5 + 1.30e5i)T^{2} \)
71 \( 1 + (-145. - 638. i)T + (-3.22e5 + 1.55e5i)T^{2} \)
73 \( 1 + (782. + 376. i)T + (2.42e5 + 3.04e5i)T^{2} \)
79 \( 1 + 1.35e3T + 4.93e5T^{2} \)
83 \( 1 + (-103. - 130. i)T + (-1.27e5 + 5.57e5i)T^{2} \)
89 \( 1 + (741. + 929. i)T + (-1.56e5 + 6.87e5i)T^{2} \)
97 \( 1 + (-88.5 + 388. i)T + (-8.22e5 - 3.95e5i)T^{2} \)
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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

−16.14290855480403655760470034849, −14.74526263580935385088445459445, −14.32550781801013795149988790225, −12.59570977471673961103329675145, −10.19262878094408588003178676675, −9.724901584115163906473053945368, −8.645221119829399040277511665162, −7.02994550316746976462904684967, −5.77311915827553988692938047546, −3.18580554473502562037295846321, 1.36218819496693602106943820304, 2.72439052532482391683113839059, 5.86374411376953324486117365380, 7.75565484557651647337064299675, 9.253152762932158372490478025814, 9.739754153000596152008157810767, 11.35516796060801281491352175634, 12.91269797971916067177592933191, 13.29351576260862339804453722958, 14.74372111262472564719378228008

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