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Properties

Label 1-485-485.309-r0-0-0
Degree $1$
Conductor $485$
Sign $0.941 + 0.337i$
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  + (0.707 + 0.707i)2-s + (−0.707 + 0.707i)3-s + i·4-s − 6-s + (−0.382 − 0.923i)7-s + (−0.707 + 0.707i)8-s − i·9-s + (0.707 − 0.707i)11-s + (−0.707 − 0.707i)12-s + (0.382 − 0.923i)13-s + (0.382 − 0.923i)14-s − 16-s + (0.382 − 0.923i)17-s + (0.707 − 0.707i)18-s + (0.382 − 0.923i)19-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + (−0.707 + 0.707i)3-s + i·4-s − 6-s + (−0.382 − 0.923i)7-s + (−0.707 + 0.707i)8-s − i·9-s + (0.707 − 0.707i)11-s + (−0.707 − 0.707i)12-s + (0.382 − 0.923i)13-s + (0.382 − 0.923i)14-s − 16-s + (0.382 − 0.923i)17-s + (0.707 − 0.707i)18-s + (0.382 − 0.923i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 485 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 485 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(485\)    =    \(5 \cdot 97\)
Sign: $0.941 + 0.337i$
Analytic conductor: \(2.25233\)
Root analytic conductor: \(2.25233\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{485} (309, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 485,\ (0:\ ),\ 0.941 + 0.337i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.355203491 + 0.2354686259i\)
\(L(\frac12)\) \(\approx\) \(1.355203491 + 0.2354686259i\)
\(L(1)\) \(\approx\) \(1.102435634 + 0.4013891458i\)
\(L(1)\) \(\approx\) \(1.102435634 + 0.4013891458i\)

Euler product

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

−23.498439370564273843598136086170, −22.75253854579442627922877694543, −22.18425186597556038170890393866, −21.34967159668436065745782995897, −20.40409352396099123151131607120, −19.09515952452855657006692884084, −19.00621465847412146096511977433, −17.99661059345497572012028885872, −16.88070801937275880943976545048, −15.96518766358605414894014091681, −14.79919977693715526965518718095, −14.13934859523071421223415770796, −12.7692438612477668575642744416, −12.5837092883439769654562479777, −11.628467935387177964166253243899, −10.93867508774212874171706446101, −9.74672580574015816968399228440, −8.88103760509890156591093753098, −7.33555655206524553054621393099, −6.27794159096116381529557428338, −5.75733121232402125747098670148, −4.62170350439562108375128517107, −3.50474821446452219034383170144, −2.10028454676313987717794555508, −1.419687209884608459317072657192, 0.70672001661440531883853513017, 3.22288246395092937114951361154, 3.68129071850451611119234405405, 4.91386566850517718971627017307, 5.60629021560136443017948669129, 6.67244664897709935481250618976, 7.36423916856630920079663590550, 8.75787442071586698555511547879, 9.6466735862515641430214095948, 10.91795066128142616686472729204, 11.49181548686664344266473876367, 12.64135227343949965821454232479, 13.517233121499377307555284989946, 14.327491350113233359008869071197, 15.404524244115053318306355481071, 16.0217105713704351790253016776, 16.86278387415747557164428198836, 17.34949431067524520012198023535, 18.37836149567855491573056687238, 19.881980662731535792314240299570, 20.66050298293364012352076433720, 21.485378462794577247112602403287, 22.450385458552895666666907631563, 22.80495148600321773582984765643, 23.62903905138246258514754893971

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