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Properties

Label 1-485-485.63-r0-0-0
Degree $1$
Conductor $485$
Sign $0.943 + 0.331i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 485 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 485 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(485\)    =    \(5 \cdot 97\)
Sign: $0.943 + 0.331i$
Analytic conductor: \(2.25233\)
Root analytic conductor: \(2.25233\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{485} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 485,\ (0:\ ),\ 0.943 + 0.331i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.644824515 + 0.2805738758i\)
\(L(\frac12)\) \(\approx\) \(1.644824515 + 0.2805738758i\)
\(L(1)\) \(\approx\) \(1.257660065 + 0.2502354533i\)
\(L(1)\) \(\approx\) \(1.257660065 + 0.2502354533i\)

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

−23.82279004698502937606216281025, −22.41139055060750451708762105112, −21.73109895990601199183205295448, −20.976125601359991099709825221472, −20.45634508402819495030438988139, −19.557731284789849333740108111323, −18.61065290863901175999070608113, −18.24364350562551336614235651677, −16.77447027269373300293622389989, −16.22349783530978027323263234209, −14.81721751468633181532968239851, −13.96323981721805613705049571841, −13.589749314787022500801069461378, −12.08170097280647984159744143090, −11.47353294554853549784175320698, −10.47206287178444668488018604725, −9.586343627246593292427305870739, −8.58631431768341002979846825218, −8.27347069536007931455132905948, −7.008592897213353228045164539786, −5.21887622092100463291344136922, −4.24005619245264271166706191481, −3.40677238086178034753002240762, −2.26182658537277614726283596729, −1.38839933966822390771084860919, 1.15669646495085168712481310927, 2.17640307211945012300742717433, 3.865230077236930575300487154806, 4.72378381989748830619844565257, 5.99679947977677375849044910665, 7.05721129285860315192118095344, 7.896625566396999050268132755995, 8.37619803805208738205862557649, 9.499845504867060576036340968328, 10.241955926107755094357495790587, 11.548967974759931284059525838916, 12.932317628537911060231894056343, 13.54304526116110596698120755181, 14.63255509016315550014667395979, 15.03731444043965067007548701449, 15.80163314765743390864404285381, 17.33648739968331985944184753507, 17.67104253605730444469463810897, 18.478467060644126079515078524727, 19.66420626974582044681471876757, 20.055535678445046968856046562579, 21.146522023432251013917907203841, 22.25675179884704218418583667545, 23.34048489480841709132886474086, 24.021467674828717116969906621620

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