Properties

Label 2-43-43.31-c1-0-1
Degree $2$
Conductor $43$
Sign $0.991 + 0.126i$
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  + (0.118 − 0.0570i)2-s + (1.09 − 0.744i)3-s + (−1.23 + 1.55i)4-s + (−1.30 − 1.21i)5-s + (0.0867 − 0.150i)6-s + (−0.00749 − 0.0129i)7-s + (−0.116 + 0.510i)8-s + (−0.458 + 1.16i)9-s + (−0.223 − 0.0689i)10-s + (−1.29 − 1.62i)11-s + (−0.195 + 2.61i)12-s + (−1.55 + 0.478i)13-s + (−0.00162 − 0.00110i)14-s + (−2.32 − 0.350i)15-s + (−0.867 − 3.79i)16-s + (4.42 − 4.10i)17-s + ⋯
L(s)  = 1  + (0.0837 − 0.0403i)2-s + (0.630 − 0.429i)3-s + (−0.618 + 0.775i)4-s + (−0.583 − 0.541i)5-s + (0.0354 − 0.0613i)6-s + (−0.00283 − 0.00490i)7-s + (−0.0411 + 0.180i)8-s + (−0.152 + 0.389i)9-s + (−0.0707 − 0.0218i)10-s + (−0.390 − 0.490i)11-s + (−0.0564 + 0.753i)12-s + (−0.430 + 0.132i)13-s + (−0.000434 − 0.000296i)14-s + (−0.600 − 0.0905i)15-s + (−0.216 − 0.949i)16-s + (1.07 − 0.995i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $0.991 + 0.126i$
Analytic conductor: \(0.343356\)
Root analytic conductor: \(0.585966\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :1/2),\ 0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.806257 - 0.0511136i\)
\(L(\frac12)\) \(\approx\) \(0.806257 - 0.0511136i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (1.84 - 6.29i)T \)
good2 \( 1 + (-0.118 + 0.0570i)T + (1.24 - 1.56i)T^{2} \)
3 \( 1 + (-1.09 + 0.744i)T + (1.09 - 2.79i)T^{2} \)
5 \( 1 + (1.30 + 1.21i)T + (0.373 + 4.98i)T^{2} \)
7 \( 1 + (0.00749 + 0.0129i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.29 + 1.62i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (1.55 - 0.478i)T + (10.7 - 7.32i)T^{2} \)
17 \( 1 + (-4.42 + 4.10i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (-2.78 - 7.09i)T + (-13.9 + 12.9i)T^{2} \)
23 \( 1 + (-5.24 + 0.790i)T + (21.9 - 6.77i)T^{2} \)
29 \( 1 + (5.92 + 4.04i)T + (10.5 + 26.9i)T^{2} \)
31 \( 1 + (-0.178 + 2.37i)T + (-30.6 - 4.62i)T^{2} \)
37 \( 1 + (2.52 - 4.36i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.20 - 1.54i)T + (25.5 - 32.0i)T^{2} \)
47 \( 1 + (6.24 - 7.83i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (6.55 + 2.02i)T + (43.7 + 29.8i)T^{2} \)
59 \( 1 + (-0.370 - 1.62i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (0.452 + 6.03i)T + (-60.3 + 9.09i)T^{2} \)
67 \( 1 + (-0.523 - 1.33i)T + (-49.1 + 45.5i)T^{2} \)
71 \( 1 + (-6.96 - 1.05i)T + (67.8 + 20.9i)T^{2} \)
73 \( 1 + (-9.32 + 2.87i)T + (60.3 - 41.1i)T^{2} \)
79 \( 1 + (6.00 + 10.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.49 + 5.10i)T + (30.3 - 77.2i)T^{2} \)
89 \( 1 + (13.1 - 8.95i)T + (32.5 - 82.8i)T^{2} \)
97 \( 1 + (-8.44 - 10.5i)T + (-21.5 + 94.5i)T^{2} \)
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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

−16.28610481168611174849308195662, −14.52562085865681315111386450794, −13.60704093189868794593026513620, −12.61519510459172113795049574619, −11.60713195499497807004302887116, −9.598031866094775375674294594350, −8.195686421005647971820426372508, −7.67993629115187433337468156032, −5.06166594297125518714009833640, −3.25274293146344409606591584418, 3.47651686551787057643016413717, 5.20809056694152857441552959087, 7.16582627639094492249646541834, 8.825667740412704769491563348694, 9.854842940417090532478385888934, 11.04669653743494269512385680742, 12.69154514403188590337864853516, 14.04565508558853176323271168477, 15.08537591799503166698948778187, 15.34587512911510912696454744443

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