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Properties

Label 1-253-253.112-r0-0-0
Degree $1$
Conductor $253$
Sign $0.557 - 0.830i$
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  + (−0.941 − 0.336i)2-s + (−0.921 + 0.389i)3-s + (0.774 + 0.633i)4-s + (−0.696 − 0.717i)5-s + (0.998 − 0.0570i)6-s + (0.993 + 0.113i)7-s + (−0.516 − 0.856i)8-s + (0.696 − 0.717i)9-s + (0.415 + 0.909i)10-s + (−0.959 − 0.281i)12-s + (−0.198 + 0.980i)13-s + (−0.897 − 0.441i)14-s + (0.921 + 0.389i)15-s + (0.198 + 0.980i)16-s + (−0.998 + 0.0570i)17-s + (−0.897 + 0.441i)18-s + ⋯
L(s)  = 1  + (−0.941 − 0.336i)2-s + (−0.921 + 0.389i)3-s + (0.774 + 0.633i)4-s + (−0.696 − 0.717i)5-s + (0.998 − 0.0570i)6-s + (0.993 + 0.113i)7-s + (−0.516 − 0.856i)8-s + (0.696 − 0.717i)9-s + (0.415 + 0.909i)10-s + (−0.959 − 0.281i)12-s + (−0.198 + 0.980i)13-s + (−0.897 − 0.441i)14-s + (0.921 + 0.389i)15-s + (0.198 + 0.980i)16-s + (−0.998 + 0.0570i)17-s + (−0.897 + 0.441i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(253\)    =    \(11 \cdot 23\)
Sign: $0.557 - 0.830i$
Analytic conductor: \(1.17492\)
Root analytic conductor: \(1.17492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{253} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 253,\ (0:\ ),\ 0.557 - 0.830i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4444939989 - 0.2368931183i\)
\(L(\frac12)\) \(\approx\) \(0.4444939989 - 0.2368931183i\)
\(L(1)\) \(\approx\) \(0.5182272202 - 0.1006617617i\)
\(L(1)\) \(\approx\) \(0.5182272202 - 0.1006617617i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.941 - 0.336i)T \)
3 \( 1 + (-0.921 + 0.389i)T \)
5 \( 1 + (-0.696 - 0.717i)T \)
7 \( 1 + (0.993 + 0.113i)T \)
13 \( 1 + (-0.198 + 0.980i)T \)
17 \( 1 + (-0.998 + 0.0570i)T \)
19 \( 1 + (-0.254 - 0.967i)T \)
29 \( 1 + (0.254 - 0.967i)T \)
31 \( 1 + (0.974 - 0.226i)T \)
37 \( 1 + (0.985 - 0.170i)T \)
41 \( 1 + (0.985 + 0.170i)T \)
43 \( 1 + (-0.654 - 0.755i)T \)
47 \( 1 + (0.309 - 0.951i)T \)
53 \( 1 + (0.870 - 0.491i)T \)
59 \( 1 + (-0.736 + 0.676i)T \)
61 \( 1 + (0.0855 - 0.996i)T \)
67 \( 1 + (0.959 - 0.281i)T \)
71 \( 1 + (0.610 - 0.791i)T \)
73 \( 1 + (-0.774 - 0.633i)T \)
79 \( 1 + (0.198 - 0.980i)T \)
83 \( 1 + (-0.466 - 0.884i)T \)
89 \( 1 + (0.654 + 0.755i)T \)
97 \( 1 + (0.466 - 0.884i)T \)
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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

−26.399333510546265958101910814634, −25.05533133759419883537112214956, −24.39592423200280171654584494406, −23.476452805057712438250868548948, −22.80696920083954302691516018594, −21.61099737995682611687253060845, −20.30817276059292898452938778340, −19.42772203043109778825623345559, −18.36688840584728303868367206650, −17.916903139696807207537642472058, −17.06641794811139975894500854194, −15.971144575343834300097863875327, −15.15546959297203412086709994958, −14.19348916919861240161809621109, −12.5325459758723422168515084638, −11.4817255829233711368393205065, −10.88313308705619770309507791640, −10.116510351996162831639829959484, −8.35119535598958533718879098240, −7.681715403714577724425305116417, −6.77643834786717579569475963274, −5.74150862754467520106058469549, −4.49775980105280364683054743291, −2.52910562671814518793344617288, −1.088817445725113206897980941313, 0.67326423472921589416673848027, 2.098591271676313581345153172922, 4.08891587291468085627514561472, 4.80049815357918061374477243463, 6.38967619538620453052273165101, 7.48728331646452397612122788776, 8.619904792077548986446316914532, 9.4048959570967624686268016556, 10.72779454380605184872874299812, 11.56125530380830791674368776318, 11.93501801997421306601394229005, 13.22473056881299988249153593499, 15.12151728979594848297226728272, 15.768094340784601104358485580415, 16.806833503395067551625068004415, 17.375297458218031068717169075000, 18.28278653884481773664015350451, 19.35904937729505657592142936505, 20.30789853052355853904269663266, 21.24089403997672871287371564020, 21.807639116285941857628085587536, 23.25341354593992338463345559032, 24.20314545604607287319258792680, 24.673171474736250712590464484596, 26.4077622229093496875205575983

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