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Properties

Label 1-407-407.137-r0-0-0
Degree $1$
Conductor $407$
Sign $0.0165 - 0.999i$
Analytic cond. $1.89010$
Root an. cond. $1.89010$
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  + (−0.104 − 0.994i)2-s + (0.669 − 0.743i)3-s + (−0.978 + 0.207i)4-s + (−0.104 + 0.994i)5-s + (−0.809 − 0.587i)6-s + (−0.978 + 0.207i)7-s + (0.309 + 0.951i)8-s + (−0.104 − 0.994i)9-s + 10-s + (−0.5 + 0.866i)12-s + (0.913 + 0.406i)13-s + (0.309 + 0.951i)14-s + (0.669 + 0.743i)15-s + (0.913 − 0.406i)16-s + (0.913 − 0.406i)17-s + (−0.978 + 0.207i)18-s + ⋯
L(s)  = 1  + (−0.104 − 0.994i)2-s + (0.669 − 0.743i)3-s + (−0.978 + 0.207i)4-s + (−0.104 + 0.994i)5-s + (−0.809 − 0.587i)6-s + (−0.978 + 0.207i)7-s + (0.309 + 0.951i)8-s + (−0.104 − 0.994i)9-s + 10-s + (−0.5 + 0.866i)12-s + (0.913 + 0.406i)13-s + (0.309 + 0.951i)14-s + (0.669 + 0.743i)15-s + (0.913 − 0.406i)16-s + (0.913 − 0.406i)17-s + (−0.978 + 0.207i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0165 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0165 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $0.0165 - 0.999i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ 0.0165 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9327493571 - 0.9174132441i\)
\(L(\frac12)\) \(\approx\) \(0.9327493571 - 0.9174132441i\)
\(L(1)\) \(\approx\) \(0.9277377161 - 0.5800642409i\)
\(L(1)\) \(\approx\) \(0.9277377161 - 0.5800642409i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.104 - 0.994i)T \)
3 \( 1 + (0.669 - 0.743i)T \)
5 \( 1 + (-0.104 + 0.994i)T \)
7 \( 1 + (-0.978 + 0.207i)T \)
13 \( 1 + (0.913 + 0.406i)T \)
17 \( 1 + (0.913 - 0.406i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 + T \)
29 \( 1 + (0.309 - 0.951i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (0.669 - 0.743i)T \)
43 \( 1 + T \)
47 \( 1 + (0.309 + 0.951i)T \)
53 \( 1 + (-0.104 - 0.994i)T \)
59 \( 1 + (0.669 + 0.743i)T \)
61 \( 1 + (-0.104 + 0.994i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.104 + 0.994i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (0.913 + 0.406i)T \)
83 \( 1 + (0.913 - 0.406i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−24.94607928412302351938901788228, −23.647476563617868166679366009932, −22.996510976186123800263940448786, −22.05626341794365510393920163362, −21.03559035527016167695042743576, −20.21939401123998349496885708278, −19.32832222160610749693782452602, −18.48269197602251682856091870930, −17.08486729894708850918148216554, −16.38144184182194249044553587539, −15.970859676806483691070185311104, −15.04313442519457840262230748983, −14.052380282560129613465768035923, −13.19714307879047459453982835815, −12.526359241149715448994761332482, −10.71003499374497464074760835415, −9.72013050445119498814701149386, −9.101353279321850284474108349068, −8.24447233773331220273119857917, −7.41156435662854801013129603323, −5.94179544773237025031568711473, −5.20338036566900844626830453827, −3.991930371309955070311387404539, −3.3232202620515947669058798487, −1.13409704901586729693119311935, 0.97413609390555590922981679164, 2.43710022068173537783642329317, 3.11019527884570374441815638545, 3.87096673712575138021399881808, 5.71889661417156259909194145926, 6.84434137741605552198735548191, 7.71809456663863276098450836664, 8.98541294887731085012077762352, 9.570308110632353446270449978365, 10.7179679330133025433490509053, 11.655424048701827949547933952992, 12.510414878156659215953790101960, 13.47499176335890421517263715111, 14.01277686462672872598618884935, 15.02958019144761256673283836673, 16.18482693691777722333583874395, 17.66053573440932185317955838125, 18.328303987238832541818183762962, 19.24957079677611243905585096414, 19.3001140312206282216239669691, 20.60434574703197389141870884508, 21.31551488169802070743291556490, 22.4887379654915172988792520635, 22.996077905304104170057098468191, 23.86903403937971519604849751565

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