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Properties

Label 1-47-47.11-r1-0-0
Degree $1$
Conductor $47$
Sign $0.755 - 0.654i$
Analytic cond. $5.05085$
Root an. cond. $5.05085$
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  + (−0.0682 − 0.997i)2-s + (0.962 + 0.269i)3-s + (−0.990 + 0.136i)4-s + (0.576 + 0.816i)5-s + (0.203 − 0.979i)6-s + (0.682 − 0.730i)7-s + (0.203 + 0.979i)8-s + (0.854 + 0.519i)9-s + (0.775 − 0.631i)10-s + (0.917 + 0.398i)11-s + (−0.990 − 0.136i)12-s + (−0.460 − 0.887i)13-s + (−0.775 − 0.631i)14-s + (0.334 + 0.942i)15-s + (0.962 − 0.269i)16-s + (−0.917 + 0.398i)17-s + ⋯
L(s)  = 1  + (−0.0682 − 0.997i)2-s + (0.962 + 0.269i)3-s + (−0.990 + 0.136i)4-s + (0.576 + 0.816i)5-s + (0.203 − 0.979i)6-s + (0.682 − 0.730i)7-s + (0.203 + 0.979i)8-s + (0.854 + 0.519i)9-s + (0.775 − 0.631i)10-s + (0.917 + 0.398i)11-s + (−0.990 − 0.136i)12-s + (−0.460 − 0.887i)13-s + (−0.775 − 0.631i)14-s + (0.334 + 0.942i)15-s + (0.962 − 0.269i)16-s + (−0.917 + 0.398i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(47\)
Sign: $0.755 - 0.654i$
Analytic conductor: \(5.05085\)
Root analytic conductor: \(5.05085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{47} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 47,\ (1:\ ),\ 0.755 - 0.654i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.959713464 - 0.7305517380i\)
\(L(\frac12)\) \(\approx\) \(1.959713464 - 0.7305517380i\)
\(L(1)\) \(\approx\) \(1.430311068 - 0.4437832778i\)
\(L(1)\) \(\approx\) \(1.430311068 - 0.4437832778i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad47 \( 1 \)
good2 \( 1 + (-0.0682 - 0.997i)T \)
3 \( 1 + (0.962 + 0.269i)T \)
5 \( 1 + (0.576 + 0.816i)T \)
7 \( 1 + (0.682 - 0.730i)T \)
11 \( 1 + (0.917 + 0.398i)T \)
13 \( 1 + (-0.460 - 0.887i)T \)
17 \( 1 + (-0.917 + 0.398i)T \)
19 \( 1 + (0.576 - 0.816i)T \)
23 \( 1 + (0.0682 - 0.997i)T \)
29 \( 1 + (-0.460 + 0.887i)T \)
31 \( 1 + (-0.962 + 0.269i)T \)
37 \( 1 + (-0.775 + 0.631i)T \)
41 \( 1 + (-0.203 + 0.979i)T \)
43 \( 1 + (0.990 - 0.136i)T \)
53 \( 1 + (0.203 - 0.979i)T \)
59 \( 1 + (-0.990 - 0.136i)T \)
61 \( 1 + (-0.775 - 0.631i)T \)
67 \( 1 + (-0.682 - 0.730i)T \)
71 \( 1 + (-0.0682 + 0.997i)T \)
73 \( 1 + (-0.854 + 0.519i)T \)
79 \( 1 + (-0.334 - 0.942i)T \)
83 \( 1 + (-0.917 - 0.398i)T \)
89 \( 1 + (-0.576 - 0.816i)T \)
97 \( 1 + (0.962 + 0.269i)T \)
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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

−33.7464728784454871090815916480, −32.78006963164321620675168616831, −31.66606332182097574962056258424, −31.05011987338525303545206684834, −29.3091187111691383995397500944, −27.779189937503387211681179541995, −26.764115269502297156218645204033, −25.446801234812582747773164476391, −24.57598336010635404615933631729, −24.155024629147212917691242134733, −22.07192500942676294144403043185, −21.00533628127084625968948032176, −19.46131699516162389577826591047, −18.23616950924207684563579831786, −17.059911284626702092571771678580, −15.68912415390687208538070048093, −14.38718324818734996176410406826, −13.62986745337475174332095065360, −12.142685278734718763711748466532, −9.339027280334829380994301391761, −8.89648236916154009247971388183, −7.45603232368048523846903166495, −5.80384167997817436707642826352, −4.23695115071085197427098314789, −1.682136473439799023858661613052, 1.773776977370820999141067036197, 3.20568889980214126295038946522, 4.6567142297319810078468390597, 7.2728171805667306519748350015, 8.86836266042677115457745174344, 10.10727989577765915707500057540, 11.04123586120247250923945169329, 12.96840506198348230497570645213, 14.12580121541342434220728939259, 14.88083015315222622543903563533, 17.28222539903704728112052228691, 18.27680656834107266287608026904, 19.77121814788066070485271928660, 20.384065160060067893139724512978, 21.74667258907658110699448544956, 22.52379550262972307932109792077, 24.40740176432459729556505758313, 25.90318104567054509793821637557, 26.80392679652062063211687862000, 27.669813417178173156157118242887, 29.386777498993244682604971757733, 30.43127003160086753589016433673, 30.81422481243076994128274685697, 32.5324642950547316287926070439, 33.18011302245490143607953981150

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