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Properties

Label 1-605-605.387-r0-0-0
Degree $1$
Conductor $605$
Sign $0.105 + 0.994i$
Analytic cond. $2.80960$
Root an. cond. $2.80960$
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  + (0.884 + 0.466i)2-s + (−0.951 − 0.309i)3-s + (0.564 + 0.825i)4-s + (−0.696 − 0.717i)6-s + (0.967 + 0.254i)7-s + (0.113 + 0.993i)8-s + (0.809 + 0.587i)9-s + (−0.281 − 0.959i)12-s + (−0.336 + 0.941i)13-s + (0.736 + 0.676i)14-s + (−0.362 + 0.931i)16-s + (0.226 − 0.974i)17-s + (0.441 + 0.897i)18-s + (0.516 + 0.856i)19-s + (−0.841 − 0.540i)21-s + ⋯
L(s)  = 1  + (0.884 + 0.466i)2-s + (−0.951 − 0.309i)3-s + (0.564 + 0.825i)4-s + (−0.696 − 0.717i)6-s + (0.967 + 0.254i)7-s + (0.113 + 0.993i)8-s + (0.809 + 0.587i)9-s + (−0.281 − 0.959i)12-s + (−0.336 + 0.941i)13-s + (0.736 + 0.676i)14-s + (−0.362 + 0.931i)16-s + (0.226 − 0.974i)17-s + (0.441 + 0.897i)18-s + (0.516 + 0.856i)19-s + (−0.841 − 0.540i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.105 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.105 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.105 + 0.994i$
Analytic conductor: \(2.80960\)
Root analytic conductor: \(2.80960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (387, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 605,\ (0:\ ),\ 0.105 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.415563769 + 1.273066951i\)
\(L(\frac12)\) \(\approx\) \(1.415563769 + 1.273066951i\)
\(L(1)\) \(\approx\) \(1.341660850 + 0.5655498976i\)
\(L(1)\) \(\approx\) \(1.341660850 + 0.5655498976i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.884 + 0.466i)T \)
3 \( 1 + (-0.951 - 0.309i)T \)
7 \( 1 + (0.967 + 0.254i)T \)
13 \( 1 + (-0.336 + 0.941i)T \)
17 \( 1 + (0.226 - 0.974i)T \)
19 \( 1 + (0.516 + 0.856i)T \)
23 \( 1 + (-0.540 - 0.841i)T \)
29 \( 1 + (-0.921 - 0.389i)T \)
31 \( 1 + (0.610 + 0.791i)T \)
37 \( 1 + (0.999 - 0.0285i)T \)
41 \( 1 + (0.985 + 0.170i)T \)
43 \( 1 + (-0.909 + 0.415i)T \)
47 \( 1 + (-0.441 + 0.897i)T \)
53 \( 1 + (-0.931 + 0.362i)T \)
59 \( 1 + (0.985 - 0.170i)T \)
61 \( 1 + (0.466 + 0.884i)T \)
67 \( 1 + (0.989 - 0.142i)T \)
71 \( 1 + (0.0855 - 0.996i)T \)
73 \( 1 + (-0.633 + 0.774i)T \)
79 \( 1 + (-0.870 + 0.491i)T \)
83 \( 1 + (-0.0570 - 0.998i)T \)
89 \( 1 + (0.654 + 0.755i)T \)
97 \( 1 + (0.980 + 0.198i)T \)
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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

−22.80530505356548925962498551988, −21.96027486269651353235877816515, −21.51834871853360738762226786574, −20.549594770891886200934782433458, −19.9178275584866998563664486664, −18.73968779777518216336015617414, −17.78047522504639951486861006086, −17.14378658645233822878117555984, −16.02691728459070664781746122837, −15.18091180936194661898682729553, −14.62093419239229268323438542702, −13.38456781707162683836038104378, −12.71497490043474911503780561181, −11.656145723696158009536161599391, −11.21570909416661943026749651703, −10.33511370938682887110281473980, −9.597367972872384112061531265841, −7.955024075158539991267700851064, −6.99165618997040536020064711582, −5.81161354509878192797126322359, −5.25293547100701593667489012993, −4.35701159066334954063473323, −3.46295073101602956958503423385, −1.99645335711302091629896250019, −0.87262141383586502610080286783, 1.49655056184961692746643632374, 2.562314216361406191167420474928, 4.16563514024465204849707846900, 4.84159251922297654962457826788, 5.66392132230264359736934871164, 6.539179232490082195888848538152, 7.46963074699156534604190739209, 8.16879537949239699878372109513, 9.616644402566604864378054979302, 10.92046400818614848796751952605, 11.69191913763683207635098420872, 12.10607150277524407320681216579, 13.12419894338071883217194733643, 14.16044147060268638668779601594, 14.64958798843470448714364323388, 15.97152449064417143623121792722, 16.421715804022049258444442552487, 17.32739650922054678659947319634, 18.1058304163078507052136640986, 18.8818617903666407839960344836, 20.299056347190930486691910000885, 21.10547412530432751390287842843, 21.76097148500701492417091713453, 22.60731203663786290525838046147, 23.2192850800954055789322511513

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