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Properties

Label 1-475-475.212-r0-0-0
Degree $1$
Conductor $475$
Sign $-0.729 + 0.684i$
Analytic cond. $2.20589$
Root an. cond. $2.20589$
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  + (0.469 + 0.882i)2-s + (−0.970 − 0.241i)3-s + (−0.559 + 0.829i)4-s + (−0.241 − 0.970i)6-s + (−0.866 − 0.5i)7-s + (−0.994 − 0.104i)8-s + (0.882 + 0.469i)9-s + (0.669 − 0.743i)11-s + (0.743 − 0.669i)12-s + (0.529 + 0.848i)13-s + (0.0348 − 0.999i)14-s + (−0.374 − 0.927i)16-s + (0.898 + 0.438i)17-s + i·18-s + (0.719 + 0.694i)21-s + (0.970 + 0.241i)22-s + ⋯
L(s)  = 1  + (0.469 + 0.882i)2-s + (−0.970 − 0.241i)3-s + (−0.559 + 0.829i)4-s + (−0.241 − 0.970i)6-s + (−0.866 − 0.5i)7-s + (−0.994 − 0.104i)8-s + (0.882 + 0.469i)9-s + (0.669 − 0.743i)11-s + (0.743 − 0.669i)12-s + (0.529 + 0.848i)13-s + (0.0348 − 0.999i)14-s + (−0.374 − 0.927i)16-s + (0.898 + 0.438i)17-s + i·18-s + (0.719 + 0.694i)21-s + (0.970 + 0.241i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.729 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.729 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.729 + 0.684i$
Analytic conductor: \(2.20589\)
Root analytic conductor: \(2.20589\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (212, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (0:\ ),\ -0.729 + 0.684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3097545025 + 0.7830971404i\)
\(L(\frac12)\) \(\approx\) \(0.3097545025 + 0.7830971404i\)
\(L(1)\) \(\approx\) \(0.7132702165 + 0.4448699772i\)
\(L(1)\) \(\approx\) \(0.7132702165 + 0.4448699772i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.469 + 0.882i)T \)
3 \( 1 + (-0.970 - 0.241i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (0.669 - 0.743i)T \)
13 \( 1 + (0.529 + 0.848i)T \)
17 \( 1 + (0.898 + 0.438i)T \)
23 \( 1 + (-0.788 + 0.615i)T \)
29 \( 1 + (0.438 + 0.898i)T \)
31 \( 1 + (-0.913 + 0.406i)T \)
37 \( 1 + (-0.951 - 0.309i)T \)
41 \( 1 + (0.374 + 0.927i)T \)
43 \( 1 + (-0.342 + 0.939i)T \)
47 \( 1 + (-0.898 + 0.438i)T \)
53 \( 1 + (0.829 + 0.559i)T \)
59 \( 1 + (0.990 + 0.139i)T \)
61 \( 1 + (-0.615 - 0.788i)T \)
67 \( 1 + (0.694 + 0.719i)T \)
71 \( 1 + (-0.961 + 0.275i)T \)
73 \( 1 + (0.529 - 0.848i)T \)
79 \( 1 + (-0.241 + 0.970i)T \)
83 \( 1 + (-0.406 - 0.913i)T \)
89 \( 1 + (-0.374 + 0.927i)T \)
97 \( 1 + (-0.694 + 0.719i)T \)
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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

−22.93354130214906233001498360779, −22.70654252319894205540805273944, −22.00944589951933608703651011383, −21.0180807590305735148397032829, −20.280449466342050064821911976435, −19.255255755904490023408318682028, −18.41538550520129379956566803827, −17.733119739001841022863660139562, −16.61392585322752469612177459107, −15.63986489097042573327633865922, −14.924418612909901603971318138461, −13.70210766617533949911976016594, −12.638344761648086281720517233083, −12.19829912094661977986546527500, −11.392232599317248010095685554512, −10.15524029411786756461000011807, −9.89070623160504627982459136829, −8.70907403232504104155644445159, −6.99383744027272941813262404902, −5.97507794008411641744391134989, −5.36048455380635815327762937411, −4.15237893547998634676417897908, −3.324801117254263870219521519701, −1.93417015472467124074749307177, −0.50844341356262609029910377244, 1.27258024559528243127760391076, 3.39186049725781220103293402003, 4.10651317671693485601435982376, 5.37008902906968046962354067202, 6.23501641261930167890250192709, 6.764415163193765033625789581375, 7.76736429518353582763810122307, 8.9762103852277741435177668364, 9.999707449327799972903586834585, 11.20353510742811785040724695430, 12.09909688036193107203498668507, 12.90433525498936263390790302634, 13.75135408463121856349544108605, 14.51702742736685847011137878967, 16.011055907656415098525772915748, 16.3125588887027869175445663037, 16.98751531931956697755401760769, 17.94639172886557938950005323281, 18.82528288198178883057462954753, 19.70135072721410374534841688806, 21.30131628537642535003794675524, 21.79189691248885581377421487468, 22.66518327184872320316829731914, 23.42458562285490930863184484980, 23.874966241151626165732762189485

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