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Properties

Label 1-245-245.114-r0-0-0
Degree $1$
Conductor $245$
Sign $0.243 - 0.969i$
Analytic cond. $1.13777$
Root an. cond. $1.13777$
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  + (0.733 − 0.680i)2-s + (0.988 − 0.149i)3-s + (0.0747 − 0.997i)4-s + (0.623 − 0.781i)6-s + (−0.623 − 0.781i)8-s + (0.955 − 0.294i)9-s + (0.955 + 0.294i)11-s + (−0.0747 − 0.997i)12-s + (0.222 + 0.974i)13-s + (−0.988 − 0.149i)16-s + (−0.826 + 0.563i)17-s + (0.5 − 0.866i)18-s + (−0.5 − 0.866i)19-s + (0.900 − 0.433i)22-s + (−0.826 − 0.563i)23-s + (−0.733 − 0.680i)24-s + ⋯
L(s)  = 1  + (0.733 − 0.680i)2-s + (0.988 − 0.149i)3-s + (0.0747 − 0.997i)4-s + (0.623 − 0.781i)6-s + (−0.623 − 0.781i)8-s + (0.955 − 0.294i)9-s + (0.955 + 0.294i)11-s + (−0.0747 − 0.997i)12-s + (0.222 + 0.974i)13-s + (−0.988 − 0.149i)16-s + (−0.826 + 0.563i)17-s + (0.5 − 0.866i)18-s + (−0.5 − 0.866i)19-s + (0.900 − 0.433i)22-s + (−0.826 − 0.563i)23-s + (−0.733 − 0.680i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.243 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.243 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.243 - 0.969i$
Analytic conductor: \(1.13777\)
Root analytic conductor: \(1.13777\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (114, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 245,\ (0:\ ),\ 0.243 - 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.906905420 - 1.487651371i\)
\(L(\frac12)\) \(\approx\) \(1.906905420 - 1.487651371i\)
\(L(1)\) \(\approx\) \(1.768439234 - 0.9083734844i\)
\(L(1)\) \(\approx\) \(1.768439234 - 0.9083734844i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.733 - 0.680i)T \)
3 \( 1 + (0.988 - 0.149i)T \)
11 \( 1 + (0.955 + 0.294i)T \)
13 \( 1 + (0.222 + 0.974i)T \)
17 \( 1 + (-0.826 + 0.563i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.826 - 0.563i)T \)
29 \( 1 + (-0.900 - 0.433i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.0747 - 0.997i)T \)
41 \( 1 + (0.623 + 0.781i)T \)
43 \( 1 + (-0.623 + 0.781i)T \)
47 \( 1 + (0.733 - 0.680i)T \)
53 \( 1 + (-0.0747 + 0.997i)T \)
59 \( 1 + (0.365 + 0.930i)T \)
61 \( 1 + (0.0747 + 0.997i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.900 + 0.433i)T \)
73 \( 1 + (0.733 + 0.680i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.222 - 0.974i)T \)
89 \( 1 + (0.955 - 0.294i)T \)
97 \( 1 - T \)
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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

−25.97244711643723591688823581173, −25.315469576231038918920877518342, −24.595291172654436339651383813483, −23.77941957229814004151319953269, −22.410956880282414348868379531345, −21.98488890520648472137089539350, −20.66358134434416418142221777133, −20.212038303088989346218233714415, −18.93387835447089904500625151696, −17.786763904542518266657168060, −16.67839755828301991885239316490, −15.70658884800913583295266561702, −14.94524053690288858665235899784, −14.095765587544605338581420824509, −13.31286370718062088068529445806, −12.3738506149225284160269965253, −11.10647472029264899996203032801, −9.61424594838743278832425404343, −8.59408725051348222543141701182, −7.77028785717033666758783279313, −6.68440952496988572331887714735, −5.498869746615907166012979698521, −4.1102301286241264119929259962, −3.417695152956618457453123618975, −2.05791150269436304128667780256, 1.57487905547955162064055758810, 2.44393545051859640992282153259, 3.881874548284877856282649425147, 4.42804139386918780109351142210, 6.227118817464063730710475237102, 7.08451713281915285984980743374, 8.77670527902052227667382230040, 9.39144949924879072917034009981, 10.63288276946320041249392209922, 11.72742231308839369341299721982, 12.742941653086013983462071569051, 13.58512253916792453949563998105, 14.46420757221061143686648802999, 15.109337420746795345180798958073, 16.2594250202686322067691030373, 17.82469422618107530256166276892, 18.88660927484010562583972180987, 19.72074359272334609402847088558, 20.20243211178733857596180144824, 21.4313557731393591748508489532, 21.890468126429417071085397658327, 23.156455653621835840115993509, 24.16810568555994224322462835164, 24.722039467734791926675659345027, 25.90864761008776765939592561532

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