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Properties

Label 1-47-47.31-r1-0-0
Degree $1$
Conductor $47$
Sign $-0.953 + 0.300i$
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  + (0.460 + 0.887i)2-s + (−0.334 + 0.942i)3-s + (−0.576 + 0.816i)4-s + (0.917 + 0.398i)5-s + (−0.990 + 0.136i)6-s + (0.854 + 0.519i)7-s + (−0.990 − 0.136i)8-s + (−0.775 − 0.631i)9-s + (0.0682 + 0.997i)10-s + (−0.962 + 0.269i)11-s + (−0.576 − 0.816i)12-s + (−0.203 − 0.979i)13-s + (−0.0682 + 0.997i)14-s + (−0.682 + 0.730i)15-s + (−0.334 − 0.942i)16-s + (0.962 + 0.269i)17-s + ⋯
L(s)  = 1  + (0.460 + 0.887i)2-s + (−0.334 + 0.942i)3-s + (−0.576 + 0.816i)4-s + (0.917 + 0.398i)5-s + (−0.990 + 0.136i)6-s + (0.854 + 0.519i)7-s + (−0.990 − 0.136i)8-s + (−0.775 − 0.631i)9-s + (0.0682 + 0.997i)10-s + (−0.962 + 0.269i)11-s + (−0.576 − 0.816i)12-s + (−0.203 − 0.979i)13-s + (−0.0682 + 0.997i)14-s + (−0.682 + 0.730i)15-s + (−0.334 − 0.942i)16-s + (0.962 + 0.269i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.953 + 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.953 + 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(47\)
Sign: $-0.953 + 0.300i$
Analytic conductor: \(5.05085\)
Root analytic conductor: \(5.05085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{47} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 47,\ (1:\ ),\ -0.953 + 0.300i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2706791052 + 1.758269982i\)
\(L(\frac12)\) \(\approx\) \(0.2706791052 + 1.758269982i\)
\(L(1)\) \(\approx\) \(0.7896398626 + 1.069991554i\)
\(L(1)\) \(\approx\) \(0.7896398626 + 1.069991554i\)

Euler product

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

−33.296660725072443682836556870155, −31.80948155476705484726593048982, −30.755607767720296258384101319239, −29.71213439769697087098797774121, −28.97031397642627463757676408420, −28.03679133564430085656082042166, −26.36191286881432928438923979919, −24.54449661367772181168982044048, −23.92303995463623569833302658023, −22.68469064837813998098683301613, −21.23138339580834041028425722901, −20.468520510080829860476536649, −18.83560753992587369031414419231, −18.01471175151146958004806430073, −16.74372036479254229986851832941, −14.23099770996834251882774454100, −13.62834440040961393327437017386, −12.36532962729688931052041155431, −11.20065942260307638739382444083, −9.81760610585220463658739876774, −7.99139893671675170666116811504, −5.99619741377964519895991173558, −4.82165329179652783949427329306, −2.36815815181982542814764062125, −1.05013691721279473794140043946, 3.07331567344334262029658302349, 5.14197486147715831827556893964, 5.670131189158640930047663469043, 7.69358768272395467704164667573, 9.28479106532600600658403404588, 10.63789844165464315009554912533, 12.3664204372937298254441690132, 14.02363224187530867765495551362, 14.99799963604229621761968738971, 16.00405472766774929548874167291, 17.5522642603088931559536197379, 18.03161326237663048457698311116, 20.76788210532382771164769895182, 21.54883922040776086320218315077, 22.46697249177646859087182784797, 23.69344245599365045930253478733, 25.13668867820734322551124206691, 26.02589080179972908948398110184, 27.159084512168229664515452112883, 28.28785735888778695750482559608, 29.86777981983487475778328307222, 31.200445762779245598668355584433, 32.31278303284865489416725029337, 33.30315335907702788283593346768, 34.09607741204914068515200288150

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