Properties

Label 2-43-43.15-c1-0-0
Degree $2$
Conductor $43$
Sign $-0.875 - 0.482i$
Analytic cond. $0.343356$
Root an. cond. $0.585966$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.72 + 2.15i)2-s + (−0.528 + 1.34i)3-s + (−1.24 − 5.47i)4-s + (0.0684 + 0.913i)5-s + (−1.99 − 3.45i)6-s + (−0.971 + 1.68i)7-s + (8.98 + 4.32i)8-s + (0.666 + 0.618i)9-s + (−2.08 − 1.42i)10-s + (−0.100 + 0.439i)11-s + (8.02 + 1.20i)12-s + (2.90 − 1.97i)13-s + (−1.95 − 4.99i)14-s + (−1.26 − 0.390i)15-s + (−14.6 + 7.06i)16-s + (0.142 − 1.90i)17-s + ⋯
L(s)  = 1  + (−1.21 + 1.52i)2-s + (−0.304 + 0.777i)3-s + (−0.624 − 2.73i)4-s + (0.0306 + 0.408i)5-s + (−0.814 − 1.41i)6-s + (−0.367 + 0.636i)7-s + (3.17 + 1.52i)8-s + (0.222 + 0.206i)9-s + (−0.660 − 0.450i)10-s + (−0.0302 + 0.132i)11-s + (2.31 + 0.349i)12-s + (0.804 − 0.548i)13-s + (−0.523 − 1.33i)14-s + (−0.326 − 0.100i)15-s + (−3.66 + 1.76i)16-s + (0.0345 − 0.461i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.875 - 0.482i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.875 - 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.875 - 0.482i$
Analytic conductor: \(0.343356\)
Root analytic conductor: \(0.585966\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :1/2),\ -0.875 - 0.482i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.104366 + 0.405777i\)
\(L(\frac12)\) \(\approx\) \(0.104366 + 0.405777i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (6.30 + 1.79i)T \)
good2 \( 1 + (1.72 - 2.15i)T + (-0.445 - 1.94i)T^{2} \)
3 \( 1 + (0.528 - 1.34i)T + (-2.19 - 2.04i)T^{2} \)
5 \( 1 + (-0.0684 - 0.913i)T + (-4.94 + 0.745i)T^{2} \)
7 \( 1 + (0.971 - 1.68i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.100 - 0.439i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (-2.90 + 1.97i)T + (4.74 - 12.1i)T^{2} \)
17 \( 1 + (-0.142 + 1.90i)T + (-16.8 - 2.53i)T^{2} \)
19 \( 1 + (-2.97 + 2.76i)T + (1.41 - 18.9i)T^{2} \)
23 \( 1 + (1.77 - 0.546i)T + (19.0 - 12.9i)T^{2} \)
29 \( 1 + (1.90 + 4.86i)T + (-21.2 + 19.7i)T^{2} \)
31 \( 1 + (-0.920 - 0.138i)T + (29.6 + 9.13i)T^{2} \)
37 \( 1 + (-0.277 - 0.480i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.111 + 0.139i)T + (-9.12 - 39.9i)T^{2} \)
47 \( 1 + (-1.72 - 7.57i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (8.52 + 5.81i)T + (19.3 + 49.3i)T^{2} \)
59 \( 1 + (-9.19 + 4.42i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (5.55 - 0.837i)T + (58.2 - 17.9i)T^{2} \)
67 \( 1 + (0.807 - 0.748i)T + (5.00 - 66.8i)T^{2} \)
71 \( 1 + (5.41 + 1.67i)T + (58.6 + 39.9i)T^{2} \)
73 \( 1 + (-11.7 + 7.99i)T + (26.6 - 67.9i)T^{2} \)
79 \( 1 + (-1.13 + 1.95i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.11 + 13.0i)T + (-60.8 - 56.4i)T^{2} \)
89 \( 1 + (-0.143 + 0.365i)T + (-65.2 - 60.5i)T^{2} \)
97 \( 1 + (3.76 - 16.4i)T + (-87.3 - 42.0i)T^{2} \)
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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

−16.21427927038395890317831303855, −15.73914121074010513060499923168, −14.90633324432490056611448786036, −13.54179009861960724491091029504, −11.07496148709720022512598190336, −10.03659159945210596841900281831, −9.127475423487938739576661217783, −7.72849731122745962061692181735, −6.32704417304187213731384989569, −5.09545607390368371911844999403, 1.32325618336817502467017257501, 3.75361341858719067122110062382, 6.98631767346689594485014415355, 8.329883497720023083103386499280, 9.571689828397463555222716363724, 10.72049647079887695160187135126, 11.88546195054389617677571281397, 12.75152470661318158717259625112, 13.60001447817664166027834476210, 16.29327540558048720698674593492

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