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Properties

Label 1-1275-1275.263-r0-0-0
Degree $1$
Conductor $1275$
Sign $-0.867 + 0.497i$
Analytic cond. $5.92107$
Root an. cond. $5.92107$
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  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (−0.707 − 0.707i)7-s + (−0.809 − 0.587i)8-s + (0.891 + 0.453i)11-s + (−0.951 − 0.309i)13-s + (0.453 − 0.891i)14-s + (0.309 − 0.951i)16-s + (0.587 − 0.809i)19-s + (−0.156 + 0.987i)22-s + (−0.453 + 0.891i)23-s − i·26-s + (0.987 + 0.156i)28-s + (−0.987 − 0.156i)29-s + (0.156 + 0.987i)31-s + 32-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (−0.707 − 0.707i)7-s + (−0.809 − 0.587i)8-s + (0.891 + 0.453i)11-s + (−0.951 − 0.309i)13-s + (0.453 − 0.891i)14-s + (0.309 − 0.951i)16-s + (0.587 − 0.809i)19-s + (−0.156 + 0.987i)22-s + (−0.453 + 0.891i)23-s − i·26-s + (0.987 + 0.156i)28-s + (−0.987 − 0.156i)29-s + (0.156 + 0.987i)31-s + 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.867 + 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.867 + 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1275\)    =    \(3 \cdot 5^{2} \cdot 17\)
Sign: $-0.867 + 0.497i$
Analytic conductor: \(5.92107\)
Root analytic conductor: \(5.92107\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1275} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1275,\ (0:\ ),\ -0.867 + 0.497i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2598375469 + 0.9749022366i\)
\(L(\frac12)\) \(\approx\) \(0.2598375469 + 0.9749022366i\)
\(L(1)\) \(\approx\) \(0.8003672987 + 0.5122075654i\)
\(L(1)\) \(\approx\) \(0.8003672987 + 0.5122075654i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.309 + 0.951i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
11 \( 1 + (0.891 + 0.453i)T \)
13 \( 1 + (-0.951 - 0.309i)T \)
19 \( 1 + (0.587 - 0.809i)T \)
23 \( 1 + (-0.453 + 0.891i)T \)
29 \( 1 + (-0.987 - 0.156i)T \)
31 \( 1 + (0.156 + 0.987i)T \)
37 \( 1 + (0.453 + 0.891i)T \)
41 \( 1 + (0.453 + 0.891i)T \)
43 \( 1 - T \)
47 \( 1 + (0.587 + 0.809i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (0.951 + 0.309i)T \)
61 \( 1 + (0.891 + 0.453i)T \)
67 \( 1 + (-0.587 + 0.809i)T \)
71 \( 1 + (-0.987 - 0.156i)T \)
73 \( 1 + (-0.891 - 0.453i)T \)
79 \( 1 + (-0.156 + 0.987i)T \)
83 \( 1 + (0.809 + 0.587i)T \)
89 \( 1 + (-0.309 - 0.951i)T \)
97 \( 1 + (-0.156 + 0.987i)T \)
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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

−20.61764959441003529410936331805, −20.00577347603788270984330947153, −19.12717155943465175145340387679, −18.79851224795787889146623187753, −17.9119596951078123822066608375, −16.87282571550687437007528431597, −16.227253055432007591401636406841, −14.98850927122271082109328160968, −14.53218230249305258420826150396, −13.657045761234351473974653980831, −12.80601699935106417020579476170, −12.04085500714331967116672040091, −11.67456122004923401843823921243, −10.55780707507484109230392554688, −9.70879952845914290297663136296, −9.20840652105707898185881133661, −8.35921715074282871308764524227, −7.09069496178803117637579002440, −6.00474872591967632599279495680, −5.47487549004853472583342529712, −4.24854419621035536000878324470, −3.57917341302948919550793875210, −2.57234456318598947569753357665, −1.82350277629671398907429957184, −0.40295337810147024565429913266, 1.1044364342370749405968901962, 2.77163359160194129514721695492, 3.66890239255719728607727736327, 4.465019494277742647751446785334, 5.33344422262031990238801414381, 6.31303622234327243014320756903, 7.12158649055591926571032910938, 7.518892670247477241763678299942, 8.693031234071521155682308031281, 9.62933751235057746324062806899, 9.98334741970991258888312499615, 11.459322930669258101191893610740, 12.21084268886393669087513721599, 13.12050802828555677084306776038, 13.633618676582876695907771414296, 14.58226206416186179639837249363, 15.12158541899804715024210426832, 16.08265992624314857851475197014, 16.66386429452878770789092797802, 17.47064606142462878306747804732, 17.88795774792668675357543733631, 19.15191388276534799648939564067, 19.76390603616094695648746949413, 20.51775210015280957228117686277, 21.76419995112487985781225639896

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