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Properties

Label 1-253-253.129-r0-0-0
Degree $1$
Conductor $253$
Sign $-0.895 + 0.445i$
Analytic cond. $1.17492$
Root an. cond. $1.17492$
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  + (0.254 + 0.967i)2-s + (−0.466 − 0.884i)3-s + (−0.870 + 0.491i)4-s + (0.564 + 0.825i)5-s + (0.736 − 0.676i)6-s + (0.0855 + 0.996i)7-s + (−0.696 − 0.717i)8-s + (−0.564 + 0.825i)9-s + (−0.654 + 0.755i)10-s + (0.841 + 0.540i)12-s + (−0.516 − 0.856i)13-s + (−0.941 + 0.336i)14-s + (0.466 − 0.884i)15-s + (0.516 − 0.856i)16-s + (−0.736 + 0.676i)17-s + (−0.941 − 0.336i)18-s + ⋯
L(s)  = 1  + (0.254 + 0.967i)2-s + (−0.466 − 0.884i)3-s + (−0.870 + 0.491i)4-s + (0.564 + 0.825i)5-s + (0.736 − 0.676i)6-s + (0.0855 + 0.996i)7-s + (−0.696 − 0.717i)8-s + (−0.564 + 0.825i)9-s + (−0.654 + 0.755i)10-s + (0.841 + 0.540i)12-s + (−0.516 − 0.856i)13-s + (−0.941 + 0.336i)14-s + (0.466 − 0.884i)15-s + (0.516 − 0.856i)16-s + (−0.736 + 0.676i)17-s + (−0.941 − 0.336i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(253\)    =    \(11 \cdot 23\)
Sign: $-0.895 + 0.445i$
Analytic conductor: \(1.17492\)
Root analytic conductor: \(1.17492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{253} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 253,\ (0:\ ),\ -0.895 + 0.445i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1920603169 + 0.8166640569i\)
\(L(\frac12)\) \(\approx\) \(0.1920603169 + 0.8166640569i\)
\(L(1)\) \(\approx\) \(0.7025452752 + 0.5320309538i\)
\(L(1)\) \(\approx\) \(0.7025452752 + 0.5320309538i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.254 + 0.967i)T \)
3 \( 1 + (-0.466 - 0.884i)T \)
5 \( 1 + (0.564 + 0.825i)T \)
7 \( 1 + (0.0855 + 0.996i)T \)
13 \( 1 + (-0.516 - 0.856i)T \)
17 \( 1 + (-0.736 + 0.676i)T \)
19 \( 1 + (0.198 + 0.980i)T \)
29 \( 1 + (-0.198 + 0.980i)T \)
31 \( 1 + (-0.985 - 0.170i)T \)
37 \( 1 + (-0.610 - 0.791i)T \)
41 \( 1 + (-0.610 + 0.791i)T \)
43 \( 1 + (-0.142 + 0.989i)T \)
47 \( 1 + (-0.809 + 0.587i)T \)
53 \( 1 + (0.921 - 0.389i)T \)
59 \( 1 + (0.974 - 0.226i)T \)
61 \( 1 + (0.897 - 0.441i)T \)
67 \( 1 + (-0.841 + 0.540i)T \)
71 \( 1 + (0.774 + 0.633i)T \)
73 \( 1 + (0.870 - 0.491i)T \)
79 \( 1 + (0.516 + 0.856i)T \)
83 \( 1 + (-0.0285 - 0.999i)T \)
89 \( 1 + (0.142 - 0.989i)T \)
97 \( 1 + (0.0285 - 0.999i)T \)
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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

−25.97063073775548171862153799876, −24.24969069671730884920392474663, −23.683384707128178058672417504850, −22.52695545707746130347757115090, −21.84711220259107640997951195463, −20.93797454579476264684819793982, −20.35335362094518611397629121674, −19.5574497202697712373014163221, −18.007985113263964903438810250104, −17.23350302312180516268340230665, −16.502164729829045095286893426305, −15.21569801745667897077333721465, −13.95452863487414067265424534277, −13.3750519981563993822431591174, −12.06473748949579975802292309394, −11.28407697126019565311624497339, −10.263768370823174221939698753570, −9.49031545266437929180681807548, −8.73248079886539066263475703303, −6.778137143393768851352604959789, −5.2882792294863444510695703418, −4.64771705268282336375291012296, −3.74766857752777236052198960735, −2.13952463510092059035605268275, −0.560046738150124289762806973594, 1.97071609772684550816768097465, 3.25533030007023221069652132530, 5.192866341897994841597899427012, 5.85738863505724375566286987207, 6.70167574516068577023962929454, 7.697486355467958621830435924182, 8.675405688634991613549961658682, 10.009554049128458728037467204274, 11.305008455812183424635155665211, 12.55335237556973459614020302038, 13.09462406012643863540463468530, 14.389944224189641204072093768300, 14.90506245859024378854578027418, 16.14504337667033311351353146337, 17.26220768834512486553305414778, 18.04188022297712123260814156524, 18.46513127631100040314758687453, 19.5752141212230515292826527417, 21.34499237525415528820145656597, 22.2677225482876644588091526345, 22.62036260357891458992956984456, 23.779703035195361962512789486597, 24.70302057517837416152760697189, 25.21517679272633884611214041254, 26.01528584827295065711238533680

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