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Properties

Label 1-43-43.26-r1-0-0
Degree $1$
Conductor $43$
Sign $0.975 + 0.222i$
Analytic cond. $4.62099$
Root an. cond. $4.62099$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.900 − 0.433i)2-s + (−0.826 + 0.563i)3-s + (0.623 − 0.781i)4-s + (0.733 + 0.680i)5-s + (−0.5 + 0.866i)6-s + (0.5 + 0.866i)7-s + (0.222 − 0.974i)8-s + (0.365 − 0.930i)9-s + (0.955 + 0.294i)10-s + (0.623 + 0.781i)11-s + (−0.0747 + 0.997i)12-s + (0.955 − 0.294i)13-s + (0.826 + 0.563i)14-s + (−0.988 − 0.149i)15-s + (−0.222 − 0.974i)16-s + (−0.733 + 0.680i)17-s + ⋯
L(s)  = 1  + (0.900 − 0.433i)2-s + (−0.826 + 0.563i)3-s + (0.623 − 0.781i)4-s + (0.733 + 0.680i)5-s + (−0.5 + 0.866i)6-s + (0.5 + 0.866i)7-s + (0.222 − 0.974i)8-s + (0.365 − 0.930i)9-s + (0.955 + 0.294i)10-s + (0.623 + 0.781i)11-s + (−0.0747 + 0.997i)12-s + (0.955 − 0.294i)13-s + (0.826 + 0.563i)14-s + (−0.988 − 0.149i)15-s + (−0.222 − 0.974i)16-s + (−0.733 + 0.680i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.975 + 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.975 + 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(43\)
Sign: $0.975 + 0.222i$
Analytic conductor: \(4.62099\)
Root analytic conductor: \(4.62099\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 43,\ (1:\ ),\ 0.975 + 0.222i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.258689823 + 0.2541167053i\)
\(L(\frac12)\) \(\approx\) \(2.258689823 + 0.2541167053i\)
\(L(1)\) \(\approx\) \(1.653192219 + 0.05796855021i\)
\(L(1)\) \(\approx\) \(1.653192219 + 0.05796855021i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 \)
good2 \( 1 + (0.900 - 0.433i)T \)
3 \( 1 + (-0.826 + 0.563i)T \)
5 \( 1 + (0.733 + 0.680i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (0.623 + 0.781i)T \)
13 \( 1 + (0.955 - 0.294i)T \)
17 \( 1 + (-0.733 + 0.680i)T \)
19 \( 1 + (-0.365 - 0.930i)T \)
23 \( 1 + (-0.988 + 0.149i)T \)
29 \( 1 + (-0.826 - 0.563i)T \)
31 \( 1 + (0.0747 - 0.997i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (-0.900 + 0.433i)T \)
47 \( 1 + (0.623 - 0.781i)T \)
53 \( 1 + (0.955 + 0.294i)T \)
59 \( 1 + (-0.222 - 0.974i)T \)
61 \( 1 + (-0.0747 - 0.997i)T \)
67 \( 1 + (0.365 + 0.930i)T \)
71 \( 1 + (0.988 + 0.149i)T \)
73 \( 1 + (-0.955 + 0.294i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.826 - 0.563i)T \)
89 \( 1 + (-0.826 + 0.563i)T \)
97 \( 1 + (0.623 + 0.781i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−33.8900892168742786431986720436, −33.31235247412913010642540410917, −32.218448652592548691537944365384, −30.651683559395297547858271020819, −29.71562555622707356279482805925, −28.89451075249490972088398854100, −27.28210562179472666117642751079, −25.558804374955164892554432512, −24.40971446970291834909985585770, −23.77892376114425939589190837341, −22.51764162259477565857262720168, −21.363002473196317329792092602664, −20.19229375195949390117936525751, −18.10521726814872836681316996556, −16.902432819337329481558410895775, −16.26830062986617721354514551180, −14.0237848617990677219434547533, −13.39625072443169823325998769451, −11.994489424331171479789826543222, −10.79513106241533262835674445188, −8.369153500266002475473558165441, −6.72980793313188962652325318123, −5.653386803915964055028845961674, −4.240448690557424757942680408184, −1.52055662773369789859442328815, 2.03374606396137481005786686155, 4.06392253663083434492089441126, 5.60309378896427297917932174750, 6.52225964408592047185018610768, 9.46065306323236572471244721715, 10.78233186465038084861458004078, 11.69632796253837196757100435867, 13.148002087742545424361940223582, 14.79340255346708243664151640424, 15.51570770159068939947454180284, 17.41781233227114921498551168685, 18.509353330912645201588174925353, 20.38603330165880432844482724559, 21.65611873848475376503043317658, 22.135438853500089263712895672962, 23.29743215507541826863534882053, 24.66185334303078595593371723066, 26.01588739306556263251305938259, 27.97595144435339470273609250614, 28.40433089672859076572935420550, 29.90744376084225968543972974436, 30.65701764388021425951281817935, 32.24754794299862457736386037118, 33.2813014252391875661133321170, 33.91462143053591610858685964814

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