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Properties

Label 1-451-451.262-r0-0-0
Degree $1$
Conductor $451$
Sign $-0.933 + 0.358i$
Analytic cond. $2.09443$
Root an. cond. $2.09443$
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  + (0.309 + 0.951i)2-s + (0.309 − 0.951i)3-s + (−0.809 + 0.587i)4-s + (−0.809 − 0.587i)5-s + 6-s + (−0.809 − 0.587i)7-s + (−0.809 − 0.587i)8-s + (−0.809 − 0.587i)9-s + (0.309 − 0.951i)10-s + (0.309 + 0.951i)12-s + (0.309 + 0.951i)13-s + (0.309 − 0.951i)14-s + (−0.809 + 0.587i)15-s + (0.309 − 0.951i)16-s + (0.309 + 0.951i)17-s + (0.309 − 0.951i)18-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + (0.309 − 0.951i)3-s + (−0.809 + 0.587i)4-s + (−0.809 − 0.587i)5-s + 6-s + (−0.809 − 0.587i)7-s + (−0.809 − 0.587i)8-s + (−0.809 − 0.587i)9-s + (0.309 − 0.951i)10-s + (0.309 + 0.951i)12-s + (0.309 + 0.951i)13-s + (0.309 − 0.951i)14-s + (−0.809 + 0.587i)15-s + (0.309 − 0.951i)16-s + (0.309 + 0.951i)17-s + (0.309 − 0.951i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 451 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.933 + 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.933 + 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(451\)    =    \(11 \cdot 41\)
Sign: $-0.933 + 0.358i$
Analytic conductor: \(2.09443\)
Root analytic conductor: \(2.09443\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{451} (262, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 451,\ (0:\ ),\ -0.933 + 0.358i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05193247903 + 0.2798388109i\)
\(L(\frac12)\) \(\approx\) \(0.05193247903 + 0.2798388109i\)
\(L(1)\) \(\approx\) \(0.7263100565 + 0.1463602673i\)
\(L(1)\) \(\approx\) \(0.7263100565 + 0.1463602673i\)

Euler product

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

−23.01004411407068641579960630244, −22.615986049906441796397729392306, −21.981839912139847974495183288240, −21.08331483146877425411241748800, −19.99640590580058347238334795135, −19.7401641118533636620554507159, −18.66810367097477573920445006130, −17.93530383019152087712474537357, −16.41234318513634399069284831190, −15.46648526942708445748085015174, −15.080505714086790664631464624358, −13.945569070830351147190377260782, −13.08105909064029401182164014468, −11.88135083778598384593311203549, −11.28226568211862685619295785690, −10.31540890570750887921191219696, −9.57057787580466596034500194789, −8.72840933683830972892576766503, −7.560046070490454414520203910009, −5.94426227296759294679270792512, −5.06996502426812662031558749570, −3.82527716100678177945670999552, −3.22243733618289167239967874027, −2.43516245539370511328799740705, −0.13966901017842494401283189918, 1.49380092710291309476865913187, 3.47185721918840001612959722654, 3.93230866777076199377188285811, 5.41670188911250368446625078614, 6.50290936271203104786450291648, 7.14600606003642151089428235577, 8.128144108332692169495699886771, 8.71296847933054546140645758525, 9.85827098427569718182935629131, 11.52296157102125450918255535456, 12.5539764340373694778416014519, 12.89004277107544395169934119757, 14.00819739181236263394206598637, 14.61450662212933358260913800047, 15.85527009580106748324006202165, 16.54984697771018299790500501356, 17.17246174861501718617871063231, 18.48444184397646429422897925970, 19.029787270302778554171695897057, 19.98497373752404557671329123748, 20.83052442987873234776523996001, 22.17244228152442947911021746018, 23.07853979328589943755570322330, 23.7040100593127881812497171105, 24.14630013288635475347683507580

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