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Properties

Label 1-7e2-49.16-r0-0-0
Degree $1$
Conductor $49$
Sign $0.991 - 0.127i$
Analytic cond. $0.227555$
Root an. cond. $0.227555$
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 + (0.365 − 0.930i)5-s + (0.623 − 0.781i)6-s + (0.623 + 0.781i)8-s + (0.955 − 0.294i)9-s + (0.365 + 0.930i)10-s + (0.955 + 0.294i)11-s + (0.0747 + 0.997i)12-s + (−0.222 − 0.974i)13-s + (−0.222 + 0.974i)15-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 + ⋯
L(s)  = 1  + (−0.733 + 0.680i)2-s + (−0.988 + 0.149i)3-s + (0.0747 − 0.997i)4-s + (0.365 − 0.930i)5-s + (0.623 − 0.781i)6-s + (0.623 + 0.781i)8-s + (0.955 − 0.294i)9-s + (0.365 + 0.930i)10-s + (0.955 + 0.294i)11-s + (0.0747 + 0.997i)12-s + (−0.222 − 0.974i)13-s + (−0.222 + 0.974i)15-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $0.991 - 0.127i$
Analytic conductor: \(0.227555\)
Root analytic conductor: \(0.227555\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 49,\ (0:\ ),\ 0.991 - 0.127i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4975307434 - 0.03194253354i\)
\(L(\frac12)\) \(\approx\) \(0.4975307434 - 0.03194253354i\)
\(L(1)\) \(\approx\) \(0.6102811862 + 0.03559613097i\)
\(L(1)\) \(\approx\) \(0.6102811862 + 0.03559613097i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
good2 \( 1 + (-0.733 + 0.680i)T \)
3 \( 1 + (-0.988 + 0.149i)T \)
5 \( 1 + (0.365 - 0.930i)T \)
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

−34.18539390577231514965402256673, −33.07393363087017020425091235792, −31.1066065814952067253425453952, −29.852326389539635081215163265833, −29.452011562929859176356143811717, −28.13168378121628631351523862342, −27.17863782032006917269889699765, −26.074369271781389671758345400655, −24.71829837801189482098432821416, −23.009283445749418963281014157, −21.99871215242018260803494001041, −21.15783088627955628892528882617, −19.17363064396215053751386493372, −18.59328269841454580966174552215, −17.24063820327101433299942508638, −16.532188310884843729015608980282, −14.47206184594732202531623351211, −12.72123722564523985542627491722, −11.51013850042803902603025228287, −10.613055392998053723355596121914, −9.370964178769244681604074071930, −7.36937270918464454681646411709, −6.18360925461122332147156635021, −3.86852105993676888812500876133, −1.78115653017227131410547689758, 1.1471600810512843307324139808, 4.79919286644587074478368981611, 5.84240042053834901489450448073, 7.29362547278191517820874441276, 9.04230079135110757187065401777, 10.08242793757184730327576037019, 11.59026146725713594453683091431, 13.07471934580024541880643963320, 14.93103467721236441065818513028, 16.21572150580853718283713191383, 17.14493338591606630769369784805, 17.83201905961061607923958775092, 19.43078038047820029136980327412, 20.78479013667969903631823792390, 22.37984779019754235614273229877, 23.53910142859883291808264941063, 24.61862782970093631143752966176, 25.50939247539995804151512450769, 27.35548018932918913598163409390, 27.77906770635638706556955703985, 28.90665059899623230325274179987, 29.89849911154759148429711258690, 32.21607122074541505699792745829, 32.79806881977478916150950057615, 33.90858231502444002250131694913

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