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Properties

Label 1-475-475.272-r1-0-0
Degree $1$
Conductor $475$
Sign $0.663 - 0.748i$
Analytic cond. $51.0458$
Root an. cond. $51.0458$
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  + (−0.694 − 0.719i)2-s + (0.927 + 0.374i)3-s + (−0.0348 + 0.999i)4-s + (−0.374 − 0.927i)6-s + (0.866 − 0.5i)7-s + (0.743 − 0.669i)8-s + (0.719 + 0.694i)9-s + (0.913 − 0.406i)11-s + (−0.406 + 0.913i)12-s + (0.970 + 0.241i)13-s + (−0.961 − 0.275i)14-s + (−0.997 − 0.0697i)16-s + (0.469 − 0.882i)17-s − i·18-s + (0.990 − 0.139i)21-s + (−0.927 − 0.374i)22-s + ⋯
L(s)  = 1  + (−0.694 − 0.719i)2-s + (0.927 + 0.374i)3-s + (−0.0348 + 0.999i)4-s + (−0.374 − 0.927i)6-s + (0.866 − 0.5i)7-s + (0.743 − 0.669i)8-s + (0.719 + 0.694i)9-s + (0.913 − 0.406i)11-s + (−0.406 + 0.913i)12-s + (0.970 + 0.241i)13-s + (−0.961 − 0.275i)14-s + (−0.997 − 0.0697i)16-s + (0.469 − 0.882i)17-s − i·18-s + (0.990 − 0.139i)21-s + (−0.927 − 0.374i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.663 - 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.663 - 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.663 - 0.748i$
Analytic conductor: \(51.0458\)
Root analytic conductor: \(51.0458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (272, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (1:\ ),\ 0.663 - 0.748i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.480375327 - 1.116499678i\)
\(L(\frac12)\) \(\approx\) \(2.480375327 - 1.116499678i\)
\(L(1)\) \(\approx\) \(1.324330651 - 0.3490890717i\)
\(L(1)\) \(\approx\) \(1.324330651 - 0.3490890717i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.694 - 0.719i)T \)
3 \( 1 + (0.927 + 0.374i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (0.913 - 0.406i)T \)
13 \( 1 + (0.970 + 0.241i)T \)
17 \( 1 + (0.469 - 0.882i)T \)
23 \( 1 + (0.829 - 0.559i)T \)
29 \( 1 + (0.882 - 0.469i)T \)
31 \( 1 + (-0.978 + 0.207i)T \)
37 \( 1 + (-0.587 - 0.809i)T \)
41 \( 1 + (-0.997 - 0.0697i)T \)
43 \( 1 + (0.342 + 0.939i)T \)
47 \( 1 + (-0.469 - 0.882i)T \)
53 \( 1 + (-0.999 - 0.0348i)T \)
59 \( 1 + (-0.438 - 0.898i)T \)
61 \( 1 + (0.559 + 0.829i)T \)
67 \( 1 + (-0.139 + 0.990i)T \)
71 \( 1 + (-0.615 - 0.788i)T \)
73 \( 1 + (-0.970 + 0.241i)T \)
79 \( 1 + (0.374 - 0.927i)T \)
83 \( 1 + (0.207 + 0.978i)T \)
89 \( 1 + (0.997 - 0.0697i)T \)
97 \( 1 + (0.139 + 0.990i)T \)
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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

−23.92550919464515113622966236933, −23.3477860969322097414343490144, −21.9904364799352428653486216291, −20.885583716473024062691179995856, −20.1871258223982674484161476419, −19.240277025435848067933322032380, −18.61538533304554300770312953560, −17.76401171828421399990915192545, −17.07270006223014170352392009282, −15.77110854352207074779459716875, −15.04662993968109136554443633451, −14.45550063973798647702458807271, −13.63182777895501029673525498893, −12.4569026566308248044852008031, −11.31754840308255542770430507858, −10.249442548232238256779142116079, −9.11886809526341469285798685613, −8.59697098436844225023296009870, −7.802154914639897746311419254264, −6.8514859388184976871549445893, −5.90295562411820378630454962381, −4.66901775622083947338199554947, −3.380547048566799667835903211514, −1.77016080181540227082100472763, −1.25141151591129004975993283375, 0.94224467416762428549125094719, 1.82622237277853919009380467541, 3.12764871068220300231301937821, 3.90754884109365780433946543473, 4.86711009675051550287572662020, 6.77788084846902454095029533069, 7.72323278384867381797527837285, 8.62160191107416135192681116271, 9.159169100056119294992132896132, 10.25910666372535220133302816808, 11.04906391492363117085032805146, 11.83693208752104687463953972909, 13.13703088827370191839915104165, 13.938406202572249193162288567105, 14.63085796962066885494751884413, 15.99735863964872233643918419199, 16.617094243202527128438661143951, 17.66571471713143878397873790219, 18.58270441221802621978248564961, 19.3030478840605637391240026519, 20.18944393165788932182783712655, 20.82376570210047657685098528719, 21.36455648667291637972048483865, 22.335323520754385979149551907, 23.45357457759108828613324740657

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