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Properties

Label 1-1275-1275.53-r0-0-0
Degree $1$
Conductor $1275$
Sign $-0.979 - 0.203i$
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  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.707 + 0.707i)7-s + (−0.309 + 0.951i)8-s + (0.156 + 0.987i)11-s + (0.587 + 0.809i)13-s + (−0.987 + 0.156i)14-s + (−0.809 + 0.587i)16-s + (−0.951 − 0.309i)19-s + (−0.453 + 0.891i)22-s + (−0.987 + 0.156i)23-s + i·26-s + (−0.891 − 0.453i)28-s + (−0.891 − 0.453i)29-s + (−0.453 − 0.891i)31-s − 32-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.707 + 0.707i)7-s + (−0.309 + 0.951i)8-s + (0.156 + 0.987i)11-s + (0.587 + 0.809i)13-s + (−0.987 + 0.156i)14-s + (−0.809 + 0.587i)16-s + (−0.951 − 0.309i)19-s + (−0.453 + 0.891i)22-s + (−0.987 + 0.156i)23-s + i·26-s + (−0.891 − 0.453i)28-s + (−0.891 − 0.453i)29-s + (−0.453 − 0.891i)31-s − 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1275\)    =    \(3 \cdot 5^{2} \cdot 17\)
Sign: $-0.979 - 0.203i$
Analytic conductor: \(5.92107\)
Root analytic conductor: \(5.92107\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1275} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1275,\ (0:\ ),\ -0.979 - 0.203i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1540464630 + 1.499119399i\)
\(L(\frac12)\) \(\approx\) \(-0.1540464630 + 1.499119399i\)
\(L(1)\) \(\approx\) \(1.000220138 + 0.8720094723i\)
\(L(1)\) \(\approx\) \(1.000220138 + 0.8720094723i\)

Euler product

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

−20.53555183374256088165553717849, −20.05673012896243056206009494815, −19.24212185245156741495324338407, −18.65204372177983790830879935674, −17.66432241817418993599981199546, −16.47118654269030505998350630326, −16.09479099315663631786883992815, −15.08564060291655870045217455613, −14.27153912289290043402120437822, −13.56499107435569050111466194469, −12.93125894001290649861816830247, −12.282768253736968206618930088120, −11.12680080347131554567366181178, −10.68533427305957423563157614349, −9.92624095442640498321140251039, −8.94756630839584405778180035226, −7.92600934297560801449932117714, −6.76910243992637493896675408251, −6.08059320695659271729396554350, −5.38320717367813653372741704367, −4.08162551766235246185962518175, −3.6128276716079933157285409077, −2.74522436392152765006552171877, −1.521801553438534338545490376493, −0.417209234727929278935427865137, 1.94437336133662439494109899198, 2.60056460699889093659462873822, 3.9222530576037436644624276999, 4.312750634650700985726845526860, 5.601082791931056899237241617510, 6.17370346616055231422237943337, 6.96151325927662787322003269760, 7.79451663339487733641920124284, 8.8606787709249695702085965, 9.43104772836351376772713371774, 10.62297000236764394541382810388, 11.742484267802100926976085484303, 12.16751306133095260415837711575, 13.12654021469377394553613687513, 13.57782481022808673997577312126, 14.80721791406777510231074484317, 15.10820267420447800722602369735, 16.01946832947351388544681539583, 16.63229150454310851960347430074, 17.45193344253768033427246042548, 18.33326081130357842098365960646, 19.11336847234423322464454186363, 20.134982995453071162462308224642, 20.78394002496444795467487496597, 21.72400805028055952990438853789

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