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Properties

Label 1-485-485.52-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: $\chi_{485} (52, \cdot )$
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

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

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