Properties

Label 2-43-43.37-c2-0-3
Degree $2$
Conductor $43$
Sign $0.640 + 0.767i$
Analytic cond. $1.17166$
Root an. cond. $1.08243$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.15i·2-s + (2.77 + 1.60i)3-s − 0.635·4-s + (−0.468 − 0.270i)5-s + (3.45 − 5.98i)6-s + (−7.68 + 4.43i)7-s − 7.24i·8-s + (0.644 + 1.11i)9-s + (−0.582 + 1.00i)10-s − 6.16·11-s + (−1.76 − 1.01i)12-s + (6.81 + 11.8i)13-s + (9.55 + 16.5i)14-s + (−0.868 − 1.50i)15-s − 18.1·16-s + (8.71 + 15.1i)17-s + ⋯
L(s)  = 1  − 1.07i·2-s + (0.925 + 0.534i)3-s − 0.158·4-s + (−0.0937 − 0.0541i)5-s + (0.575 − 0.996i)6-s + (−1.09 + 0.633i)7-s − 0.905i·8-s + (0.0715 + 0.123i)9-s + (−0.0582 + 0.100i)10-s − 0.560·11-s + (−0.147 − 0.0849i)12-s + (0.524 + 0.908i)13-s + (0.682 + 1.18i)14-s + (−0.0578 − 0.100i)15-s − 1.13·16-s + (0.512 + 0.888i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.640 + 0.767i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.640 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.640 + 0.767i$
Analytic conductor: \(1.17166\)
Root analytic conductor: \(1.08243\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :1),\ 0.640 + 0.767i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.20689 - 0.564524i\)
\(L(\frac12)\) \(\approx\) \(1.20689 - 0.564524i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-30.5 + 30.3i)T \)
good2 \( 1 + 2.15iT - 4T^{2} \)
3 \( 1 + (-2.77 - 1.60i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (0.468 + 0.270i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (7.68 - 4.43i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + 6.16T + 121T^{2} \)
13 \( 1 + (-6.81 - 11.8i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (-8.71 - 15.1i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (1.57 + 0.912i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-12.8 + 22.2i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-17.4 + 10.0i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (27.1 - 47.0i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-8.14 - 4.70i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 48.7T + 1.68e3T^{2} \)
47 \( 1 + 56.1T + 2.20e3T^{2} \)
53 \( 1 + (-27.2 + 47.1i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 - 28.4T + 3.48e3T^{2} \)
61 \( 1 + (28.7 - 16.5i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-1.46 + 2.54i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (102. - 58.9i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-112. + 64.9i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-59.5 - 103. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (46.8 - 81.2i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (127. + 73.7i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 66.7T + 9.40e3T^{2} \)
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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

−15.63971598429679273357463521911, −14.36480506018532032708356003041, −12.96713227657228327294781032593, −12.10867232707208552749961845530, −10.61037052729238759763150754036, −9.602017276054941061227449750597, −8.596099335342314917933289410680, −6.44587416223577296971031506720, −3.85142171110210299077940350168, −2.63807099224144248612488579381, 3.05048057921254500990940456716, 5.70591089279072168371953428010, 7.26275808855017233869105248852, 7.88192419390825148559171641644, 9.400459613703445597411576082639, 11.05146526584975901921004400154, 13.00376228916113041503504941821, 13.66766312648611497057894994796, 14.85212891461712045366029878085, 15.85906437412010340344068115931

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