Properties

Label 1-605-605.234-r0-0-0
Degree $1$
Conductor $605$
Sign $0.999 + 0.0311i$
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  + (0.985 + 0.170i)2-s + (0.809 − 0.587i)3-s + (0.941 + 0.336i)4-s + (0.897 − 0.441i)6-s + (0.362 + 0.931i)7-s + (0.870 + 0.491i)8-s + (0.309 − 0.951i)9-s + (0.959 − 0.281i)12-s + (0.0285 − 0.999i)13-s + (0.198 + 0.980i)14-s + (0.774 + 0.633i)16-s + (−0.516 + 0.856i)17-s + (0.466 − 0.884i)18-s + (0.0855 − 0.996i)19-s + (0.841 + 0.540i)21-s + ⋯
L(s)  = 1  + (0.985 + 0.170i)2-s + (0.809 − 0.587i)3-s + (0.941 + 0.336i)4-s + (0.897 − 0.441i)6-s + (0.362 + 0.931i)7-s + (0.870 + 0.491i)8-s + (0.309 − 0.951i)9-s + (0.959 − 0.281i)12-s + (0.0285 − 0.999i)13-s + (0.198 + 0.980i)14-s + (0.774 + 0.633i)16-s + (−0.516 + 0.856i)17-s + (0.466 − 0.884i)18-s + (0.0855 − 0.996i)19-s + (0.841 + 0.540i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0311i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0311i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.999 + 0.0311i$
Analytic conductor: \(2.80960\)
Root analytic conductor: \(2.80960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (234, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 605,\ (0:\ ),\ 0.999 + 0.0311i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.613221411 - 0.05629184215i\)
\(L(\frac12)\) \(\approx\) \(3.613221411 - 0.05629184215i\)
\(L(1)\) \(\approx\) \(2.489627666 + 0.02580965937i\)
\(L(1)\) \(\approx\) \(2.489627666 + 0.02580965937i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.985 - 0.170i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 + (-0.362 - 0.931i)T \)
13 \( 1 + (-0.0285 + 0.999i)T \)
17 \( 1 + (0.516 - 0.856i)T \)
19 \( 1 + (-0.0855 + 0.996i)T \)
23 \( 1 + (0.841 - 0.540i)T \)
29 \( 1 + (-0.974 + 0.226i)T \)
31 \( 1 + (0.564 - 0.825i)T \)
37 \( 1 + (0.610 + 0.791i)T \)
41 \( 1 + (-0.696 - 0.717i)T \)
43 \( 1 + (0.415 + 0.909i)T \)
47 \( 1 + (-0.466 - 0.884i)T \)
53 \( 1 + (0.774 - 0.633i)T \)
59 \( 1 + (-0.696 + 0.717i)T \)
61 \( 1 + (0.985 - 0.170i)T \)
67 \( 1 + (-0.142 - 0.989i)T \)
71 \( 1 + (0.921 + 0.389i)T \)
73 \( 1 + (-0.998 - 0.0570i)T \)
79 \( 1 + (0.736 + 0.676i)T \)
83 \( 1 + (-0.254 + 0.967i)T \)
89 \( 1 + (0.654 + 0.755i)T \)
97 \( 1 + (0.993 - 0.113i)T \)
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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

−22.924178046628590652018048734, −22.22543302099812424221195959763, −21.28999432274352716603143967832, −20.69650420540572425057065202609, −20.09899072260384799247794962957, −19.33299929661455468244677995599, −18.306607395265395487074397815122, −16.76590387848252864562018653669, −16.28276395581974236246418234691, −15.3879158260746763291402862122, −14.30395337449786651855560281635, −14.06601224675666331261772031975, −13.272319145017841510091828247368, −12.09053032530888992983614599088, −11.17143892559673981745714667055, −10.344759183538489244893333659060, −9.57693783069685314391491228207, −8.28755940707650832937355660009, −7.37925452287946237589777570415, −6.484640096761383098852751593679, −5.100553102679835722964718566792, −4.29520668697338781281222121061, −3.70940276527797456975086112678, −2.51365272563088554991474380810, −1.55753004524066391073552064869, 1.57629808705632793697180311386, 2.51007732092703416702596762519, 3.28687526115483110436928774889, 4.441331098507106861124571550724, 5.57732560247077943016424410568, 6.364713589875868196920421971312, 7.42938956173539112758187896284, 8.22429738699935874390556484585, 9.01263853105446548304582684842, 10.41578282716234167377446229596, 11.48186746718173205766577093222, 12.4148346225526891118493151262, 12.90525495439569815090149222926, 13.861438913448359814296015103, 14.58602154094281278956918239193, 15.460963356282280538433718985707, 15.7905518105160374066219880589, 17.49954521864785915571825331406, 17.93183140677732382611152915054, 19.25460829791732151094432938319, 19.84643269365514592654772825518, 20.63410361157414149326221764851, 21.59553969921488392881158352061, 22.047070153449651759308596191129, 23.264780976668927544448367556960

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