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Properties

Label 1-605-605.49-r0-0-0
Degree $1$
Conductor $605$
Sign $0.279 - 0.960i$
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  + (−0.696 − 0.717i)2-s + (−0.309 − 0.951i)3-s + (−0.0285 + 0.999i)4-s + (−0.466 + 0.884i)6-s + (−0.774 + 0.633i)7-s + (0.736 − 0.676i)8-s + (−0.809 + 0.587i)9-s + (0.959 − 0.281i)12-s + (−0.610 + 0.791i)13-s + (0.993 + 0.113i)14-s + (−0.998 − 0.0570i)16-s + (−0.0855 − 0.996i)17-s + (0.985 + 0.170i)18-s + (−0.921 − 0.389i)19-s + (0.841 + 0.540i)21-s + ⋯
L(s)  = 1  + (−0.696 − 0.717i)2-s + (−0.309 − 0.951i)3-s + (−0.0285 + 0.999i)4-s + (−0.466 + 0.884i)6-s + (−0.774 + 0.633i)7-s + (0.736 − 0.676i)8-s + (−0.809 + 0.587i)9-s + (0.959 − 0.281i)12-s + (−0.610 + 0.791i)13-s + (0.993 + 0.113i)14-s + (−0.998 − 0.0570i)16-s + (−0.0855 − 0.996i)17-s + (0.985 + 0.170i)18-s + (−0.921 − 0.389i)19-s + (0.841 + 0.540i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.279 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.279 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.279 - 0.960i$
Analytic conductor: \(2.80960\)
Root analytic conductor: \(2.80960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 605,\ (0:\ ),\ 0.279 - 0.960i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4761590642 - 0.3574136870i\)
\(L(\frac12)\) \(\approx\) \(0.4761590642 - 0.3574136870i\)
\(L(1)\) \(\approx\) \(0.5192298215 - 0.2629217264i\)
\(L(1)\) \(\approx\) \(0.5192298215 - 0.2629217264i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.696 - 0.717i)T \)
3 \( 1 + (-0.309 - 0.951i)T \)
7 \( 1 + (-0.774 + 0.633i)T \)
13 \( 1 + (-0.610 + 0.791i)T \)
17 \( 1 + (-0.0855 - 0.996i)T \)
19 \( 1 + (-0.921 - 0.389i)T \)
23 \( 1 + (-0.841 + 0.540i)T \)
29 \( 1 + (0.516 + 0.856i)T \)
31 \( 1 + (0.941 - 0.336i)T \)
37 \( 1 + (0.564 - 0.825i)T \)
41 \( 1 + (0.897 - 0.441i)T \)
43 \( 1 + (-0.415 - 0.909i)T \)
47 \( 1 + (0.985 - 0.170i)T \)
53 \( 1 + (0.998 - 0.0570i)T \)
59 \( 1 + (0.897 + 0.441i)T \)
61 \( 1 + (0.696 - 0.717i)T \)
67 \( 1 + (0.142 + 0.989i)T \)
71 \( 1 + (0.974 - 0.226i)T \)
73 \( 1 + (0.254 + 0.967i)T \)
79 \( 1 + (0.198 + 0.980i)T \)
83 \( 1 + (0.362 + 0.931i)T \)
89 \( 1 + (-0.654 - 0.755i)T \)
97 \( 1 + (0.870 + 0.491i)T \)
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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

−23.160397699752301855616301660691, −22.66191169548400335923850139457, −21.71139178879851633961767717292, −20.618367432873885565709347744270, −19.77636684091403347249939934212, −19.243400563449788464619823890347, −17.932363166499917233314368670322, −17.19344435202247294572240105997, −16.66133373758626793798298347459, −15.83738756422574723194178657544, −15.097178562703041230717709653938, −14.40042336840244226852640219484, −13.26848975004733019572503679899, −12.12595126562378863559612363705, −10.826844718173425298192825380567, −10.1892411886354850968795344, −9.76128885383495114237617054259, −8.556814057624120004472859689244, −7.82973039573129659176577679722, −6.43038914948735854254235816961, −6.036414568987670553319918195774, −4.75392579647375091341376559578, −3.94714800794186865293566660624, −2.54043050512899500282226166910, −0.69476885746038762894817995367, 0.65738920412392859664721121676, 2.184918161426823884656569609772, 2.60626067304730920347169594754, 4.04383103926721391359325211075, 5.40978629696280632680982381040, 6.65836773521329785761326920637, 7.20960339820017359386343740222, 8.37257113629969686186604804524, 9.1416397098096715350592398908, 10.016774346451983669839580504493, 11.15639278760996982105088939314, 11.9429787089980902787100907994, 12.464916125761775474646978194672, 13.34256769679042431868230753434, 14.18439364516339397275008644780, 15.69400852159719583525303672062, 16.49860870995005439446385419389, 17.30887754578965913573266602779, 18.125395550948068621203326474784, 18.81285262297093844806001363892, 19.46850635282339793289162329327, 20.03387515847798404595505181654, 21.31060902659751643980460256221, 22.02404564914417288269148404937, 22.73847437429562634061936068443

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