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Properties

Label 1-731-731.12-r0-0-0
Degree $1$
Conductor $731$
Sign $0.520 - 0.853i$
Analytic cond. $3.39474$
Root an. cond. $3.39474$
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  + (−0.111 − 0.993i)2-s + (0.720 + 0.693i)3-s + (−0.974 + 0.222i)4-s + (0.312 − 0.949i)5-s + (0.608 − 0.793i)6-s + (0.130 − 0.991i)7-s + (0.330 + 0.943i)8-s + (0.0373 + 0.999i)9-s + (−0.978 − 0.204i)10-s + (0.985 − 0.167i)11-s + (−0.856 − 0.516i)12-s + (0.563 + 0.826i)13-s + (−0.999 − 0.0186i)14-s + (0.884 − 0.467i)15-s + (0.900 − 0.433i)16-s + ⋯
L(s)  = 1  + (−0.111 − 0.993i)2-s + (0.720 + 0.693i)3-s + (−0.974 + 0.222i)4-s + (0.312 − 0.949i)5-s + (0.608 − 0.793i)6-s + (0.130 − 0.991i)7-s + (0.330 + 0.943i)8-s + (0.0373 + 0.999i)9-s + (−0.978 − 0.204i)10-s + (0.985 − 0.167i)11-s + (−0.856 − 0.516i)12-s + (0.563 + 0.826i)13-s + (−0.999 − 0.0186i)14-s + (0.884 − 0.467i)15-s + (0.900 − 0.433i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.520 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.520 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.520 - 0.853i$
Analytic conductor: \(3.39474\)
Root analytic conductor: \(3.39474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 731,\ (0:\ ),\ 0.520 - 0.853i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.630167652 - 0.9151638177i\)
\(L(\frac12)\) \(\approx\) \(1.630167652 - 0.9151638177i\)
\(L(1)\) \(\approx\) \(1.248828255 - 0.5151442840i\)
\(L(1)\) \(\approx\) \(1.248828255 - 0.5151442840i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
43 \( 1 \)
good2 \( 1 + (-0.111 - 0.993i)T \)
3 \( 1 + (0.720 + 0.693i)T \)
5 \( 1 + (0.312 - 0.949i)T \)
7 \( 1 + (0.130 - 0.991i)T \)
11 \( 1 + (0.985 - 0.167i)T \)
13 \( 1 + (0.563 + 0.826i)T \)
19 \( 1 + (-0.0373 + 0.999i)T \)
23 \( 1 + (0.638 + 0.770i)T \)
29 \( 1 + (-0.0186 + 0.999i)T \)
31 \( 1 + (0.516 - 0.856i)T \)
37 \( 1 + (0.991 - 0.130i)T \)
41 \( 1 + (0.276 - 0.960i)T \)
47 \( 1 + (0.974 - 0.222i)T \)
53 \( 1 + (-0.982 + 0.185i)T \)
59 \( 1 + (-0.943 - 0.330i)T \)
61 \( 1 + (-0.856 + 0.516i)T \)
67 \( 1 + (0.733 - 0.680i)T \)
71 \( 1 + (-0.638 + 0.770i)T \)
73 \( 1 + (-0.978 + 0.204i)T \)
79 \( 1 + (0.991 + 0.130i)T \)
83 \( 1 + (-0.399 - 0.916i)T \)
89 \( 1 + (0.930 + 0.365i)T \)
97 \( 1 + (0.578 - 0.815i)T \)
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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

−22.776691824280301894496445600845, −22.05780068017476959353117608116, −21.222017805751677353602941817064, −19.93591231154772969668802726614, −19.0949350194638380242030158733, −18.526891201843686579696658744882, −17.80664420698442643228434017288, −17.277009367924566342948423641864, −15.807083721691146004906110676588, −15.06543383718206052757871670342, −14.67467868172921730009009080885, −13.78644900405778821737983814058, −13.064161165111033053584381881275, −12.110988637062257601278544188887, −10.954248092625738589688713781727, −9.67360539505011512351541236474, −9.03512578033855781360407281823, −8.226856599496098126659709453252, −7.364713228878780790332759622042, −6.3680644024962357569125903082, −6.09666439840468194011122010494, −4.65552370050733240039306197525, −3.343031652407849024001621965903, −2.50962375716373203094609344811, −1.142319834632737443955557345896, 1.170755908096940692992557758610, 1.85223237936997522406175334518, 3.3461466705259590297464215459, 4.092644789427489540335618558977, 4.61270931171398290861699188611, 5.84413646933913318405826126186, 7.480166356599557870408436042817, 8.43613646449480268908775448284, 9.180235934960695551494414368356, 9.68690121040542648980375192390, 10.6738487183853891911142558325, 11.43307377300623115424043898926, 12.47067363212143785490841557851, 13.488178356592712074206016459694, 13.916011051745502574631203676, 14.66417027805084527791348828717, 16.106460950693214241523224814477, 16.84460391218726461339105467979, 17.29286049051386064255159396524, 18.67675056607803651866722475705, 19.4463956593658177794279520761, 20.18336592619328071347192956169, 20.65932182220922035235456780800, 21.33655564433424420444125221420, 22.01429085115161248331192032278

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