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Properties

Label 1-7e2-49.20-r1-0-0
Degree $1$
Conductor $49$
Sign $0.740 + 0.672i$
Analytic cond. $5.26578$
Root an. cond. $5.26578$
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  + (0.623 + 0.781i)2-s + (0.900 − 0.433i)3-s + (−0.222 + 0.974i)4-s + (0.900 − 0.433i)5-s + (0.900 + 0.433i)6-s + (−0.900 + 0.433i)8-s + (0.623 − 0.781i)9-s + (0.900 + 0.433i)10-s + (0.623 + 0.781i)11-s + (0.222 + 0.974i)12-s + (−0.623 − 0.781i)13-s + (0.623 − 0.781i)15-s + (−0.900 − 0.433i)16-s + (0.222 + 0.974i)17-s + 18-s − 19-s + ⋯
L(s)  = 1  + (0.623 + 0.781i)2-s + (0.900 − 0.433i)3-s + (−0.222 + 0.974i)4-s + (0.900 − 0.433i)5-s + (0.900 + 0.433i)6-s + (−0.900 + 0.433i)8-s + (0.623 − 0.781i)9-s + (0.900 + 0.433i)10-s + (0.623 + 0.781i)11-s + (0.222 + 0.974i)12-s + (−0.623 − 0.781i)13-s + (0.623 − 0.781i)15-s + (−0.900 − 0.433i)16-s + (0.222 + 0.974i)17-s + 18-s − 19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.740 + 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.740 + 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $0.740 + 0.672i$
Analytic conductor: \(5.26578\)
Root analytic conductor: \(5.26578\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 49,\ (1:\ ),\ 0.740 + 0.672i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.711655742 + 1.047561695i\)
\(L(\frac12)\) \(\approx\) \(2.711655742 + 1.047561695i\)
\(L(1)\) \(\approx\) \(1.933522022 + 0.5738591870i\)
\(L(1)\) \(\approx\) \(1.933522022 + 0.5738591870i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
good2 \( 1 + (0.623 + 0.781i)T \)
3 \( 1 + (0.900 - 0.433i)T \)
5 \( 1 + (0.900 - 0.433i)T \)
11 \( 1 + (0.623 + 0.781i)T \)
13 \( 1 + (-0.623 - 0.781i)T \)
17 \( 1 + (0.222 + 0.974i)T \)
19 \( 1 - T \)
23 \( 1 + (-0.222 + 0.974i)T \)
29 \( 1 + (-0.222 - 0.974i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.222 - 0.974i)T \)
41 \( 1 + (0.900 - 0.433i)T \)
43 \( 1 + (-0.900 - 0.433i)T \)
47 \( 1 + (-0.623 - 0.781i)T \)
53 \( 1 + (-0.222 + 0.974i)T \)
59 \( 1 + (0.900 + 0.433i)T \)
61 \( 1 + (0.222 + 0.974i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.222 + 0.974i)T \)
73 \( 1 + (-0.623 + 0.781i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.623 + 0.781i)T \)
89 \( 1 + (-0.623 + 0.781i)T \)
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

−33.01560139295338333624237958665, −32.1585639030322079227528821369, −31.17853261586234143853442850195, −29.96572782187551974006290702295, −29.235582475837828427823084306708, −27.6491798946677525058519066472, −26.5921531224977351772189947951, −25.23095736517513893733739845866, −24.16045647030041821108524918740, −22.32777274430256496607072567858, −21.63298590356526447955803139245, −20.67080485422128521366183663117, −19.40876700595867945637524705065, −18.45506147970721820399419395353, −16.480519454042238572028363106059, −14.62161504329244519139432522169, −14.18847625044676389670844948985, −12.94147629736310746170412329113, −11.17148374710816495178862994010, −9.91626169415771365371355080226, −8.95429599466715683624038432701, −6.5796101135264379510338645468, −4.8227586467431939975791017115, −3.24437793919478916539078001260, −1.96323991504187471942015450856, 2.13236234001519963484114991641, 3.999754848446052884324849861623, 5.74996501742913203102593225399, 7.16641085502735283874832682399, 8.49668518900253679599271469358, 9.73285854943002944160326683542, 12.41914299114577662934123956293, 13.139893037680417360428596609547, 14.40443193621918326798585861609, 15.23411472443455792064336297697, 17.04334488761349903628423445064, 17.85393097478600669665774001469, 19.68545494722767291326778875470, 20.92051115222391578605172456948, 21.93014023991814398008288745123, 23.46352958972426754175324641674, 24.67043890390799255253186107425, 25.330422012179091899533939523108, 26.1706684691188098121106605213, 27.72983058329608430044130366313, 29.65931081922095993237119792697, 30.31869346408141019386676508977, 31.681459882926586810158890675026, 32.46733635813848866039795340910, 33.352536735586069059405344435

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