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Properties

Label 1-417-417.50-r0-0-0
Degree $1$
Conductor $417$
Sign $0.0709 + 0.997i$
Analytic cond. $1.93653$
Root an. cond. $1.93653$
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  + (0.0227 − 0.999i)2-s + (−0.998 − 0.0455i)4-s + (−0.377 + 0.926i)5-s + (−0.419 − 0.907i)7-s + (−0.0682 + 0.997i)8-s + (0.917 + 0.398i)10-s + (0.158 − 0.987i)11-s + (0.113 + 0.993i)13-s + (−0.917 + 0.398i)14-s + (0.995 + 0.0909i)16-s + (−0.648 − 0.761i)17-s + (−0.983 + 0.181i)19-s + (0.419 − 0.907i)20-s + (−0.983 − 0.181i)22-s + (−0.576 + 0.816i)23-s + ⋯
L(s)  = 1  + (0.0227 − 0.999i)2-s + (−0.998 − 0.0455i)4-s + (−0.377 + 0.926i)5-s + (−0.419 − 0.907i)7-s + (−0.0682 + 0.997i)8-s + (0.917 + 0.398i)10-s + (0.158 − 0.987i)11-s + (0.113 + 0.993i)13-s + (−0.917 + 0.398i)14-s + (0.995 + 0.0909i)16-s + (−0.648 − 0.761i)17-s + (−0.983 + 0.181i)19-s + (0.419 − 0.907i)20-s + (−0.983 − 0.181i)22-s + (−0.576 + 0.816i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 417 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0709 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 417 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0709 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(417\)    =    \(3 \cdot 139\)
Sign: $0.0709 + 0.997i$
Analytic conductor: \(1.93653\)
Root analytic conductor: \(1.93653\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{417} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 417,\ (0:\ ),\ 0.0709 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1974797562 + 0.1839309578i\)
\(L(\frac12)\) \(\approx\) \(0.1974797562 + 0.1839309578i\)
\(L(1)\) \(\approx\) \(0.6210370820 - 0.2051501738i\)
\(L(1)\) \(\approx\) \(0.6210370820 - 0.2051501738i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
139 \( 1 \)
good2 \( 1 + (0.0227 - 0.999i)T \)
5 \( 1 + (-0.377 + 0.926i)T \)
7 \( 1 + (-0.419 - 0.907i)T \)
11 \( 1 + (0.158 - 0.987i)T \)
13 \( 1 + (0.113 + 0.993i)T \)
17 \( 1 + (-0.648 - 0.761i)T \)
19 \( 1 + (-0.983 + 0.181i)T \)
23 \( 1 + (-0.576 + 0.816i)T \)
29 \( 1 + (-0.538 + 0.842i)T \)
31 \( 1 + (0.291 + 0.956i)T \)
37 \( 1 + (-0.247 + 0.968i)T \)
41 \( 1 + (-0.746 - 0.665i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (-0.613 + 0.789i)T \)
53 \( 1 + (-0.877 + 0.480i)T \)
59 \( 1 + (0.460 - 0.887i)T \)
61 \( 1 + (-0.746 + 0.665i)T \)
67 \( 1 + (0.934 + 0.356i)T \)
71 \( 1 + (-0.613 - 0.789i)T \)
73 \( 1 + (-0.898 + 0.439i)T \)
79 \( 1 + (0.203 + 0.979i)T \)
83 \( 1 + (-0.934 + 0.356i)T \)
89 \( 1 + (0.715 - 0.699i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−24.26839248923351968557753386193, −23.21965964012535028908845615701, −22.56412125038936805790748408853, −21.686739691535093474894002987585, −20.558042485372723436593002220049, −19.61006927457334893134591176579, −18.72036122305809537472771016336, −17.655018894197332840451746158524, −17.06067376316685052580833085284, −16.00834920875708447150803132919, −15.293876379431623999841872594, −14.81743774236524392971759485481, −13.167989883166806407048886426948, −12.80886190515747354419443166965, −11.93271492035240023490721292573, −10.297404495367074405594671666340, −9.30371639916825747085093704586, −8.517056041859928457673746863610, −7.81536191345673874743978584485, −6.51019200911346249204219217771, −5.695539359133223798423092227971, −4.67349737003300418409493764880, −3.83054660410765857386435084147, −2.08867072110947670528052203832, −0.154468124626484145634574367572, 1.554829640454840482235561383523, 2.945772239828487857799060166787, 3.68963199989656514823156628461, 4.58667900234176532911061169145, 6.182585220288742776433798504209, 7.09986513966493558619191261625, 8.32319323837626824880346872640, 9.360569650609053368392668487174, 10.35740505825227097894909372588, 11.10466063486518408151591126908, 11.689036846768346435092439390770, 12.94888430411068627064845671124, 13.94853039347622399906373514286, 14.26290787328424943726920459070, 15.7168238404214437080482684703, 16.71590911093048674432233918141, 17.71135980554553558885182094893, 18.71377017288226767421870646366, 19.28509921376589711772249410218, 19.96186324942764437860766713123, 21.033773090115313666305263182203, 21.90291986546612912315269155570, 22.51916773074952698497228194370, 23.498147634455857875609139697906, 23.955732700782081143988655002915

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