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Properties

Label 1-247-247.176-r1-0-0
Degree $1$
Conductor $247$
Sign $-0.109 - 0.993i$
Analytic cond. $26.5438$
Root an. cond. $26.5438$
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  + (0.342 − 0.939i)2-s + (0.173 + 0.984i)3-s + (−0.766 − 0.642i)4-s + (−0.984 + 0.173i)5-s + (0.984 + 0.173i)6-s + (−0.866 + 0.5i)7-s + (−0.866 + 0.5i)8-s + (−0.939 + 0.342i)9-s + (−0.173 + 0.984i)10-s + (0.866 + 0.5i)11-s + (0.5 − 0.866i)12-s + (0.173 + 0.984i)14-s + (−0.342 − 0.939i)15-s + (0.173 + 0.984i)16-s + (−0.173 − 0.984i)17-s + i·18-s + ⋯
L(s)  = 1  + (0.342 − 0.939i)2-s + (0.173 + 0.984i)3-s + (−0.766 − 0.642i)4-s + (−0.984 + 0.173i)5-s + (0.984 + 0.173i)6-s + (−0.866 + 0.5i)7-s + (−0.866 + 0.5i)8-s + (−0.939 + 0.342i)9-s + (−0.173 + 0.984i)10-s + (0.866 + 0.5i)11-s + (0.5 − 0.866i)12-s + (0.173 + 0.984i)14-s + (−0.342 − 0.939i)15-s + (0.173 + 0.984i)16-s + (−0.173 − 0.984i)17-s + i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 247 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.109 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 247 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.109 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(247\)    =    \(13 \cdot 19\)
Sign: $-0.109 - 0.993i$
Analytic conductor: \(26.5438\)
Root analytic conductor: \(26.5438\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{247} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 247,\ (1:\ ),\ -0.109 - 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6348955231 - 0.7087135350i\)
\(L(\frac12)\) \(\approx\) \(0.6348955231 - 0.7087135350i\)
\(L(1)\) \(\approx\) \(0.8436288412 - 0.1861495821i\)
\(L(1)\) \(\approx\) \(0.8436288412 - 0.1861495821i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.342 - 0.939i)T \)
3 \( 1 + (0.173 + 0.984i)T \)
5 \( 1 + (-0.984 + 0.173i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
17 \( 1 + (-0.173 - 0.984i)T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (0.173 - 0.984i)T \)
31 \( 1 + (-0.866 - 0.5i)T \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + (-0.984 + 0.173i)T \)
43 \( 1 + (0.939 + 0.342i)T \)
47 \( 1 + (0.642 - 0.766i)T \)
53 \( 1 + (0.173 - 0.984i)T \)
59 \( 1 + (0.984 - 0.173i)T \)
61 \( 1 + (0.766 + 0.642i)T \)
67 \( 1 + (0.984 + 0.173i)T \)
71 \( 1 + (0.342 - 0.939i)T \)
73 \( 1 + (-0.342 + 0.939i)T \)
79 \( 1 + (-0.939 - 0.342i)T \)
83 \( 1 + (-0.866 - 0.5i)T \)
89 \( 1 + (0.642 - 0.766i)T \)
97 \( 1 + (-0.984 + 0.173i)T \)
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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

−25.87383042283201754134603798911, −25.100396040313885775516806558143, −24.106344563557391948585155422036, −23.57079980253060209202561557982, −22.80081289941526976757705469741, −21.91387810745797562616490770218, −20.30645630460429432350919122778, −19.36607434430908359353185657547, −18.835419267567341902157262194610, −17.42772949350240738796255689188, −16.75249867614162823590594474900, −15.79790532726981089646041346174, −14.719931740150802380099787359931, −13.86552729320902242527413864649, −12.8067483240095054360362286627, −12.30364354788434107575035648919, −10.99763594027199716849833263174, −9.09789807931741667008661625770, −8.44004021758396584337329450276, −7.21722317939569650696362848222, −6.76576483826060918526964014346, −5.56985950473485312422826299599, −3.94208425243574683755731015082, −3.24241430171530977482047366607, −0.957782154288711028012062069674, 0.34577055759424095691009047770, 2.51019315762973998228268085014, 3.46821047887791507876136154823, 4.27481773058959682316237227490, 5.3511805695713712621226726899, 6.82006947695195251236568608992, 8.57939433049901732848134505716, 9.354804028662618183577773530204, 10.21128520400729991994632144033, 11.407685802210170982071667328177, 11.93963296417764919331674347509, 13.12712105670930832497388254293, 14.40128925414160210028166422016, 15.16119413085243127745044029033, 15.948175540516812447578749773734, 17.1295695120081333283418253664, 18.61305334455043012137853048587, 19.384139688384577837711487229057, 20.12282228159861789472850631664, 20.87797585223766910927636946970, 22.13751775645148902240396140186, 22.55737276558670576262229311823, 23.21645824802741728074632750890, 24.6711296325929739907079437021, 25.82900068034747704505319357544

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