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Properties

Label 1-35e2-1225.1123-r0-0-0
Degree $1$
Conductor $1225$
Sign $0.879 + 0.475i$
Analytic cond. $5.68887$
Root an. cond. $5.68887$
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  + (−0.0598 − 0.998i)2-s + (−0.850 − 0.525i)3-s + (−0.992 + 0.119i)4-s + (−0.473 + 0.880i)6-s + (0.178 + 0.983i)8-s + (0.447 + 0.894i)9-s + (−0.447 + 0.894i)11-s + (0.907 + 0.420i)12-s + (0.351 − 0.936i)13-s + (0.971 − 0.237i)16-s + (−0.800 − 0.599i)17-s + (0.866 − 0.5i)18-s + (−0.104 − 0.994i)19-s + (0.919 + 0.393i)22-s + (0.817 + 0.575i)23-s + (0.365 − 0.930i)24-s + ⋯
L(s)  = 1  + (−0.0598 − 0.998i)2-s + (−0.850 − 0.525i)3-s + (−0.992 + 0.119i)4-s + (−0.473 + 0.880i)6-s + (0.178 + 0.983i)8-s + (0.447 + 0.894i)9-s + (−0.447 + 0.894i)11-s + (0.907 + 0.420i)12-s + (0.351 − 0.936i)13-s + (0.971 − 0.237i)16-s + (−0.800 − 0.599i)17-s + (0.866 − 0.5i)18-s + (−0.104 − 0.994i)19-s + (0.919 + 0.393i)22-s + (0.817 + 0.575i)23-s + (0.365 − 0.930i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.879 + 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.879 + 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $0.879 + 0.475i$
Analytic conductor: \(5.68887\)
Root analytic conductor: \(5.68887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (1123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (0:\ ),\ 0.879 + 0.475i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3381325431 + 0.08550942858i\)
\(L(\frac12)\) \(\approx\) \(0.3381325431 + 0.08550942858i\)
\(L(1)\) \(\approx\) \(0.5105532134 - 0.3022306972i\)
\(L(1)\) \(\approx\) \(0.5105532134 - 0.3022306972i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.0598 - 0.998i)T \)
3 \( 1 + (-0.850 - 0.525i)T \)
11 \( 1 + (-0.447 + 0.894i)T \)
13 \( 1 + (0.351 - 0.936i)T \)
17 \( 1 + (-0.800 - 0.599i)T \)
19 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + (0.817 + 0.575i)T \)
29 \( 1 + (-0.753 + 0.657i)T \)
31 \( 1 + (-0.913 - 0.406i)T \)
37 \( 1 + (-0.907 - 0.420i)T \)
41 \( 1 + (-0.134 + 0.990i)T \)
43 \( 1 + (-0.433 - 0.900i)T \)
47 \( 1 + (0.967 + 0.251i)T \)
53 \( 1 + (0.119 + 0.992i)T \)
59 \( 1 + (-0.280 + 0.959i)T \)
61 \( 1 + (0.873 - 0.486i)T \)
67 \( 1 + (0.406 - 0.913i)T \)
71 \( 1 + (-0.393 + 0.919i)T \)
73 \( 1 + (-0.635 + 0.772i)T \)
79 \( 1 + (-0.913 + 0.406i)T \)
83 \( 1 + (-0.266 + 0.963i)T \)
89 \( 1 + (-0.163 + 0.986i)T \)
97 \( 1 + (0.587 - 0.809i)T \)
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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

−21.32799627204809158941187143720, −20.57199649328678941995358385993, −19.00465550929318433260119937581, −18.76036576163206005518598300144, −17.764333512017225657284466850, −17.04156901825110695108933042146, −16.41862341425484539684028449067, −15.913992591666828946914640621817, −15.05444206511699683817947886777, −14.34436112883650216134081262892, −13.36371594130619709212285555038, −12.67619513873517023106289297710, −11.595983632520330702520636329224, −10.7682931129226863354397104357, −10.07588967466650864790508368147, −8.99262755519940443698768768523, −8.5358498601689370821139131862, −7.30742352079636868093750399929, −6.485172731051554071012826310229, −5.85908835012604558293591545539, −5.08611028548227502167892022749, −4.15632473627091713172779345045, −3.48183348262728184642654632475, −1.619430548318460633186547203726, −0.19462652079288760997460625278, 1.00703381467779184795786066496, 2.04024696995692519045264298559, 2.875087355988952610435874933871, 4.12329351359790040929894848499, 5.08282469719837939239756741988, 5.540880313216858209507134312415, 6.96904739216444069719425635269, 7.54873575804044555955759184432, 8.6961479204155764854619622979, 9.53048152293071537807582441083, 10.53704865363043902933203868920, 11.016286628237601820488928478663, 11.74021148539505290495541053273, 12.7439348564739532673075967558, 13.03573058072671871122228147932, 13.8008475752651051398103740154, 15.07486877411850999324523504493, 15.79542988063179253200353942524, 17.04315152558681804264132156169, 17.52920710468200178452379325118, 18.254782007295058878617613371321, 18.69573861359031802721627590381, 19.857712882117452444188496628344, 20.22772550900959179140871332972, 21.19915401680431059402977554413

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