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Properties

Label 1-485-485.117-r0-0-0
Degree $1$
Conductor $485$
Sign $0.295 - 0.955i$
Analytic cond. $2.25233$
Root an. cond. $2.25233$
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  + (−0.382 − 0.923i)2-s + (0.923 + 0.382i)3-s + (−0.707 + 0.707i)4-s − i·6-s + (−0.980 − 0.195i)7-s + (0.923 + 0.382i)8-s + (0.707 + 0.707i)9-s + (0.382 − 0.923i)11-s + (−0.923 + 0.382i)12-s + (−0.195 − 0.980i)13-s + (0.195 + 0.980i)14-s − i·16-s + (0.195 + 0.980i)17-s + (0.382 − 0.923i)18-s + (−0.195 − 0.980i)19-s + ⋯
L(s)  = 1  + (−0.382 − 0.923i)2-s + (0.923 + 0.382i)3-s + (−0.707 + 0.707i)4-s − i·6-s + (−0.980 − 0.195i)7-s + (0.923 + 0.382i)8-s + (0.707 + 0.707i)9-s + (0.382 − 0.923i)11-s + (−0.923 + 0.382i)12-s + (−0.195 − 0.980i)13-s + (0.195 + 0.980i)14-s − i·16-s + (0.195 + 0.980i)17-s + (0.382 − 0.923i)18-s + (−0.195 − 0.980i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 485 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.295 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 485 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.295 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(485\)    =    \(5 \cdot 97\)
Sign: $0.295 - 0.955i$
Analytic conductor: \(2.25233\)
Root analytic conductor: \(2.25233\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{485} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 485,\ (0:\ ),\ 0.295 - 0.955i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.054026604 - 0.7770556209i\)
\(L(\frac12)\) \(\approx\) \(1.054026604 - 0.7770556209i\)
\(L(1)\) \(\approx\) \(0.9930857521 - 0.4033268664i\)
\(L(1)\) \(\approx\) \(0.9930857521 - 0.4033268664i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
97 \( 1 \)
good2 \( 1 + (-0.382 - 0.923i)T \)
3 \( 1 + (0.923 + 0.382i)T \)
7 \( 1 + (-0.980 - 0.195i)T \)
11 \( 1 + (0.382 - 0.923i)T \)
13 \( 1 + (-0.195 - 0.980i)T \)
17 \( 1 + (0.195 + 0.980i)T \)
19 \( 1 + (-0.195 - 0.980i)T \)
23 \( 1 + (0.831 + 0.555i)T \)
29 \( 1 + (0.555 - 0.831i)T \)
31 \( 1 + (0.923 + 0.382i)T \)
37 \( 1 + (-0.555 - 0.831i)T \)
41 \( 1 + (0.831 + 0.555i)T \)
43 \( 1 + (-0.707 - 0.707i)T \)
47 \( 1 + (-0.707 - 0.707i)T \)
53 \( 1 + (0.923 - 0.382i)T \)
59 \( 1 + (-0.555 - 0.831i)T \)
61 \( 1 + T \)
67 \( 1 + (0.195 + 0.980i)T \)
71 \( 1 + (0.831 - 0.555i)T \)
73 \( 1 + (0.707 - 0.707i)T \)
79 \( 1 + (0.923 + 0.382i)T \)
83 \( 1 + (0.980 - 0.195i)T \)
89 \( 1 + (-0.382 + 0.923i)T \)
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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

−24.250651317845462988699436475865, −23.07951088319650168100746619950, −22.6672571095655578478821304112, −21.3322394846225319407505670050, −20.28038689909865592239705980690, −19.40968952391717631717615396869, −18.826528808137781464306451342564, −18.11224285943622724052160239500, −16.946949722482200001967108337560, −16.17738234927462643561334106112, −15.29429614536717579901677522802, −14.489659970715699840027526991289, −13.825780355283756330147038142319, −12.83498811003098868773801219907, −12.020617268543415605410303668844, −10.214683372539802874676503718656, −9.50499511843288003002112343312, −8.92235656330240223483483666337, −7.8436482586347165151681824991, −6.84264062633387968320014765343, −6.492402302277269482053958025894, −4.90829104464436732216044723140, −3.882913074505705118128933062122, −2.56547586483662373997106958002, −1.25463934055543513388969368811, 0.86475284485447817032959386196, 2.4136188141806456075873735545, 3.26686839296317250983357413141, 3.87194512307963421869624940131, 5.18831902885796584090174640502, 6.7271092905165803713305281459, 7.96589644187573834786946841228, 8.680904744617751762430035854399, 9.53688771275252522847579292769, 10.29931330704264134282047989343, 11.05016787714367994523311233868, 12.36474672368121624393619183228, 13.24372966896769949242573096349, 13.69576871916852852671750797167, 14.948543071652627028392642694738, 15.870881964179791483321560594, 16.83224047157912387062121021118, 17.68390450489234671270543009296, 19.0065525283852796975797661907, 19.47176034342173176137072115122, 19.90351255631323903337348005342, 21.07806869876662918409501120481, 21.62644112409040219173489810687, 22.41906422158145084649803342161, 23.338345196820414358841584060867

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