Properties

Label 2-43-43.13-c1-0-0
Degree $2$
Conductor $43$
Sign $-0.169 - 0.985i$
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  + (−0.483 + 2.11i)2-s + (−0.240 + 0.222i)3-s + (−2.45 − 1.18i)4-s + (0.0260 + 0.00392i)5-s + (−0.356 − 0.616i)6-s + (1.56 − 2.71i)7-s + (0.984 − 1.23i)8-s + (−0.216 + 2.88i)9-s + (−0.0208 + 0.0532i)10-s + (3.36 − 1.62i)11-s + (0.853 − 0.263i)12-s + (−1.78 − 4.55i)13-s + (4.98 + 4.62i)14-s + (−0.00712 + 0.00485i)15-s + (−1.25 − 1.57i)16-s + (−5.84 + 0.881i)17-s + ⋯
L(s)  = 1  + (−0.342 + 1.49i)2-s + (−0.138 + 0.128i)3-s + (−1.22 − 0.591i)4-s + (0.0116 + 0.00175i)5-s + (−0.145 − 0.251i)6-s + (0.591 − 1.02i)7-s + (0.348 − 0.436i)8-s + (−0.0720 + 0.961i)9-s + (−0.00660 + 0.0168i)10-s + (1.01 − 0.488i)11-s + (0.246 − 0.0760i)12-s + (−0.496 − 1.26i)13-s + (1.33 + 1.23i)14-s + (−0.00183 + 0.00125i)15-s + (−0.314 − 0.394i)16-s + (−1.41 + 0.213i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43\)
Sign: $-0.169 - 0.985i$
Analytic conductor: \(0.343356\)
Root analytic conductor: \(0.585966\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 43,\ (\ :1/2),\ -0.169 - 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.424898 + 0.504471i\)
\(L(\frac12)\) \(\approx\) \(0.424898 + 0.504471i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 + (-6.54 + 0.341i)T \)
good2 \( 1 + (0.483 - 2.11i)T + (-1.80 - 0.867i)T^{2} \)
3 \( 1 + (0.240 - 0.222i)T + (0.224 - 2.99i)T^{2} \)
5 \( 1 + (-0.0260 - 0.00392i)T + (4.77 + 1.47i)T^{2} \)
7 \( 1 + (-1.56 + 2.71i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.36 + 1.62i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (1.78 + 4.55i)T + (-9.52 + 8.84i)T^{2} \)
17 \( 1 + (5.84 - 0.881i)T + (16.2 - 5.01i)T^{2} \)
19 \( 1 + (-0.296 - 3.95i)T + (-18.7 + 2.83i)T^{2} \)
23 \( 1 + (-0.284 - 0.193i)T + (8.40 + 21.4i)T^{2} \)
29 \( 1 + (-0.971 - 0.901i)T + (2.16 + 28.9i)T^{2} \)
31 \( 1 + (-2.06 + 0.638i)T + (25.6 - 17.4i)T^{2} \)
37 \( 1 + (-5.01 - 8.69i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.431 - 1.89i)T + (-36.9 - 17.7i)T^{2} \)
47 \( 1 + (10.0 + 4.82i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (2.12 - 5.41i)T + (-38.8 - 36.0i)T^{2} \)
59 \( 1 + (-0.107 - 0.134i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (-5.31 - 1.63i)T + (50.4 + 34.3i)T^{2} \)
67 \( 1 + (0.0822 + 1.09i)T + (-66.2 + 9.98i)T^{2} \)
71 \( 1 + (4.37 - 2.98i)T + (25.9 - 66.0i)T^{2} \)
73 \( 1 + (2.70 + 6.88i)T + (-53.5 + 49.6i)T^{2} \)
79 \( 1 + (-2.70 + 4.68i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.49 - 1.38i)T + (6.20 - 82.7i)T^{2} \)
89 \( 1 + (-2.93 + 2.72i)T + (6.65 - 88.7i)T^{2} \)
97 \( 1 + (8.11 - 3.90i)T + (60.4 - 75.8i)T^{2} \)
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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

−16.47971891751574482271534630441, −15.33408216153283057883264273465, −14.28533390759817113197388777809, −13.42290768950974482724241736018, −11.38598978150519041666466092450, −10.09642815100240970325828759190, −8.396734749976490527397574152036, −7.56331372349205409795934001001, −6.13201291478136233402654405737, −4.61792356286841438820273108404, 2.14658750650857661649073908884, 4.29120729359600041872843454959, 6.61441496055964137251381488756, 9.009732890816113394393154452052, 9.425772754906663838916920454948, 11.43552282252419398297077696695, 11.69777779256258156414088205696, 12.82747367250164263272577826983, 14.40634159276742343817091985371, 15.56566394026028095318945337804

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