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Properties

Label 1-605-605.299-r1-0-0
Degree $1$
Conductor $605$
Sign $0.276 - 0.960i$
Analytic cond. $65.0162$
Root an. cond. $65.0162$
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  + (0.941 + 0.336i)2-s + (−0.309 + 0.951i)3-s + (0.774 + 0.633i)4-s + (−0.610 + 0.791i)6-s + (−0.736 + 0.676i)7-s + (0.516 + 0.856i)8-s + (−0.809 − 0.587i)9-s + (−0.841 + 0.540i)12-s + (−0.998 − 0.0570i)13-s + (−0.921 + 0.389i)14-s + (0.198 + 0.980i)16-s + (−0.466 − 0.884i)17-s + (−0.564 − 0.825i)18-s + (0.985 + 0.170i)19-s + (−0.415 − 0.909i)21-s + ⋯
L(s)  = 1  + (0.941 + 0.336i)2-s + (−0.309 + 0.951i)3-s + (0.774 + 0.633i)4-s + (−0.610 + 0.791i)6-s + (−0.736 + 0.676i)7-s + (0.516 + 0.856i)8-s + (−0.809 − 0.587i)9-s + (−0.841 + 0.540i)12-s + (−0.998 − 0.0570i)13-s + (−0.921 + 0.389i)14-s + (0.198 + 0.980i)16-s + (−0.466 − 0.884i)17-s + (−0.564 − 0.825i)18-s + (0.985 + 0.170i)19-s + (−0.415 − 0.909i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.276 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.276 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.276 - 0.960i$
Analytic conductor: \(65.0162\)
Root analytic conductor: \(65.0162\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 605,\ (1:\ ),\ 0.276 - 0.960i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01573947450 + 0.01184631307i\)
\(L(\frac12)\) \(\approx\) \(0.01573947450 + 0.01184631307i\)
\(L(1)\) \(\approx\) \(0.9822582539 + 0.6846845062i\)
\(L(1)\) \(\approx\) \(0.9822582539 + 0.6846845062i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.941 + 0.336i)T \)
3 \( 1 + (-0.309 + 0.951i)T \)
7 \( 1 + (-0.736 + 0.676i)T \)
13 \( 1 + (-0.998 - 0.0570i)T \)
17 \( 1 + (-0.466 - 0.884i)T \)
19 \( 1 + (0.985 + 0.170i)T \)
23 \( 1 + (-0.415 + 0.909i)T \)
29 \( 1 + (-0.897 + 0.441i)T \)
31 \( 1 + (-0.362 - 0.931i)T \)
37 \( 1 + (0.254 - 0.967i)T \)
41 \( 1 + (0.0285 - 0.999i)T \)
43 \( 1 + (-0.654 + 0.755i)T \)
47 \( 1 + (0.564 - 0.825i)T \)
53 \( 1 + (-0.198 + 0.980i)T \)
59 \( 1 + (-0.0285 - 0.999i)T \)
61 \( 1 + (-0.941 + 0.336i)T \)
67 \( 1 + (0.959 - 0.281i)T \)
71 \( 1 + (0.696 + 0.717i)T \)
73 \( 1 + (0.993 + 0.113i)T \)
79 \( 1 + (-0.0855 - 0.996i)T \)
83 \( 1 + (-0.870 - 0.491i)T \)
89 \( 1 + (-0.142 + 0.989i)T \)
97 \( 1 + (-0.974 + 0.226i)T \)
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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

−22.41864073666703948921440002091, −21.690874257145575664546511457373, −20.28442508456009828784087334271, −19.87989003918871765430391443374, −19.13574470064114823851607380376, −18.23665699106028170842677767631, −17.03881205414325323356777280505, −16.47461370147608434397656372661, −15.32734115872193691015503335120, −14.32107923662913860229676607302, −13.63227144531102829724979956272, −12.834521253793080284708254642880, −12.289845926820894017456394960983, −11.34403063663875030318207360262, −10.47687251454165855865374173881, −9.57348018083652921372769545975, −8.00550018036099460045646990518, −7.039068028483927217452203472196, −6.48376685727532431517212878848, −5.49078514909538827785745308859, −4.47060821093663529763209290065, −3.297254237894651255477045674643, −2.33761690835934519743058612596, −1.22441815890773727867446343414, −0.00337125855389569945256745548, 2.29626193132615115483327115871, 3.20770356402022907605525771050, 4.06594212320273536634320300886, 5.26320479653855061736510997492, 5.613550757799232106644536121659, 6.78986027168733744312041038958, 7.736124619350885754608194983536, 9.174050823795681563036037858528, 9.71800640437162475404284332757, 10.99084216863955295199139198188, 11.78977079435632511296333834309, 12.43721251465763864318032080477, 13.52949825001401198524021271127, 14.43808123936478993053876320114, 15.28013751141330831194256717103, 15.850938580215977485991656089012, 16.54767288458262018002288369281, 17.37377903122510688097456496823, 18.45303563650359111903119099954, 19.872902464626916845237198764125, 20.2989196687277684245715872279, 21.43045555525743749435746530031, 22.03757873646453750569444160757, 22.500876711884303775795256308052, 23.2443454261706447877119247023

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