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Properties

Label 1-485-485.28-r0-0-0
Degree $1$
Conductor $485$
Sign $-0.891 + 0.453i$
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 + (−0.195 − 0.980i)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 + (0.980 + 0.195i)13-s + (−0.980 − 0.195i)14-s + i·16-s + (−0.980 − 0.195i)17-s + (−0.382 − 0.923i)18-s + (0.980 + 0.195i)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 + (−0.195 − 0.980i)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 + (0.980 + 0.195i)13-s + (−0.980 − 0.195i)14-s + i·16-s + (−0.980 − 0.195i)17-s + (−0.382 − 0.923i)18-s + (0.980 + 0.195i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 485 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 485 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(485\)    =    \(5 \cdot 97\)
Sign: $-0.891 + 0.453i$
Analytic conductor: \(2.25233\)
Root analytic conductor: \(2.25233\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{485} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 485,\ (0:\ ),\ -0.891 + 0.453i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1337713194 - 0.5584714084i\)
\(L(\frac12)\) \(\approx\) \(-0.1337713194 - 0.5584714084i\)
\(L(1)\) \(\approx\) \(0.5572118216 - 0.4858324319i\)
\(L(1)\) \(\approx\) \(0.5572118216 - 0.4858324319i\)

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 \( 1 + (-0.195 - 0.980i)T \)
11 \( 1 + (-0.382 - 0.923i)T \)
13 \( 1 + (0.980 + 0.195i)T \)
17 \( 1 + (-0.980 - 0.195i)T \)
19 \( 1 + (0.980 + 0.195i)T \)
23 \( 1 + (-0.555 - 0.831i)T \)
29 \( 1 + (0.831 - 0.555i)T \)
31 \( 1 + (-0.923 + 0.382i)T \)
37 \( 1 + (-0.831 - 0.555i)T \)
41 \( 1 + (-0.555 - 0.831i)T \)
43 \( 1 + (-0.707 + 0.707i)T \)
47 \( 1 + (-0.707 + 0.707i)T \)
53 \( 1 + (-0.923 - 0.382i)T \)
59 \( 1 + (-0.831 - 0.555i)T \)
61 \( 1 + T \)
67 \( 1 + (-0.980 - 0.195i)T \)
71 \( 1 + (-0.555 + 0.831i)T \)
73 \( 1 + (0.707 + 0.707i)T \)
79 \( 1 + (-0.923 + 0.382i)T \)
83 \( 1 + (0.195 - 0.980i)T \)
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

−24.044631692246392278104012580181, −23.5405294091156628380617309102, −22.56659927830806059441818396429, −22.1069044310946394789201678989, −21.265493706541979430684336244271, −20.01056276036699314367639905528, −18.59611412411970467777511371001, −18.07071427756220014059222914070, −17.544423122729391577338058987964, −16.33588830344129173041115141504, −15.69268109196966430688082752618, −15.12372566804636715548677531765, −13.67297043886863680620863330180, −13.05528164451243122363286989317, −12.1847509683086928744307086524, −11.47410615154238217826064230046, −10.12878918632922749388700138155, −9.04208724406668483464096063497, −8.032632971707358871913243545152, −7.035721022242887618139335203565, −6.24661868650966318277020375242, −5.40455070284777627067503948306, −4.6845781434412366519704822485, −3.299988908298202234653717507478, −1.77311974663575027321183042675, 0.32435492856821058590491222736, 1.48042999088806864618569723697, 3.20309585465908522295703981776, 3.99438719147930555385758745012, 4.90926300254011418629574460002, 5.93885696158811902014110276425, 6.769452798168850180235459655605, 8.3726532148497172617828814257, 9.48562802663880585966994782140, 10.43722626636696070961875275786, 10.97772695573916447645973610251, 11.680620531319380589288915154, 12.78734020346080569139592096575, 13.57574127277161992377996076021, 14.29972507804157748044413504764, 15.80270373951938599604745471693, 16.23309476014006949157925188675, 17.49235346290541785633477079625, 18.18018845574758823894378710011, 19.02125334977402438021224939161, 20.165078793331233583341572561233, 20.76683404409846017688379195579, 21.61983643766934446091248070916, 22.419543289607035664510755460681, 23.06496527978647612908857725718

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