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Properties

Label 1-39e2-1521.1121-r1-0-0
Degree $1$
Conductor $1521$
Sign $-0.645 - 0.763i$
Analytic cond. $163.454$
Root an. cond. $163.454$
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  + (−0.987 + 0.160i)2-s + (0.948 − 0.316i)4-s + (0.919 − 0.391i)5-s + (0.885 − 0.464i)7-s + (−0.885 + 0.464i)8-s + (−0.845 + 0.534i)10-s + (0.632 − 0.774i)11-s + (−0.799 + 0.600i)14-s + (0.799 − 0.600i)16-s + (0.845 + 0.534i)17-s + (−0.5 − 0.866i)19-s + (0.748 − 0.663i)20-s + (−0.5 + 0.866i)22-s − 23-s + (0.692 − 0.721i)25-s + ⋯
L(s)  = 1  + (−0.987 + 0.160i)2-s + (0.948 − 0.316i)4-s + (0.919 − 0.391i)5-s + (0.885 − 0.464i)7-s + (−0.885 + 0.464i)8-s + (−0.845 + 0.534i)10-s + (0.632 − 0.774i)11-s + (−0.799 + 0.600i)14-s + (0.799 − 0.600i)16-s + (0.845 + 0.534i)17-s + (−0.5 − 0.866i)19-s + (0.748 − 0.663i)20-s + (−0.5 + 0.866i)22-s − 23-s + (0.692 − 0.721i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.645 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.645 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $-0.645 - 0.763i$
Analytic conductor: \(163.454\)
Root analytic conductor: \(163.454\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1521} (1121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1521,\ (1:\ ),\ -0.645 - 0.763i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5968681808 - 1.286921103i\)
\(L(\frac12)\) \(\approx\) \(0.5968681808 - 1.286921103i\)
\(L(1)\) \(\approx\) \(0.8731784652 - 0.2219404724i\)
\(L(1)\) \(\approx\) \(0.8731784652 - 0.2219404724i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.987 + 0.160i)T \)
5 \( 1 + (0.919 - 0.391i)T \)
7 \( 1 + (0.885 - 0.464i)T \)
11 \( 1 + (0.632 - 0.774i)T \)
17 \( 1 + (0.845 + 0.534i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.987 + 0.160i)T \)
31 \( 1 + (0.692 + 0.721i)T \)
37 \( 1 + (0.278 - 0.960i)T \)
41 \( 1 + (-0.568 - 0.822i)T \)
43 \( 1 + (-0.970 - 0.239i)T \)
47 \( 1 + (0.200 + 0.979i)T \)
53 \( 1 + (-0.885 + 0.464i)T \)
59 \( 1 + (-0.799 - 0.600i)T \)
61 \( 1 + (0.885 + 0.464i)T \)
67 \( 1 + (-0.748 + 0.663i)T \)
71 \( 1 + (-0.428 + 0.903i)T \)
73 \( 1 + (-0.354 - 0.935i)T \)
79 \( 1 + (-0.200 - 0.979i)T \)
83 \( 1 + (-0.428 - 0.903i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.120 + 0.992i)T \)
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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

−20.640244609705889616058191374677, −20.01924421395887289040064489169, −18.80411109713420815710830420168, −18.4636867469366054807470792811, −17.79407576309780010447823501054, −17.01605045350553078617535459909, −16.61064922940073685331590221445, −15.29310047979293107480175118075, −14.80093033401806944751069038310, −14.08086253515508016713135767569, −12.97199928896207129094470668206, −11.93257958489370540740643760662, −11.60517241950364561360790946651, −10.4901618629658618387197221059, −9.85106168884580561911082846099, −9.34279662785692907340769718839, −8.26456170831212356640192790775, −7.71017469946035609130666648282, −6.656111802413550113394296865866, −6.010509784058460851170138394998, −5.06702720355365174687286591944, −3.80289711475032303627755325154, −2.65593720352787495415142304656, −1.84740895410030754676541045642, −1.314156826964306049328555553270, 0.3408490077906825970540806936, 1.35848247988041980791584961233, 1.87766820104003554085220571302, 3.08734035044826174483230059306, 4.31602052662649727819300404518, 5.45001452943480086080588874554, 6.05511808432699190821943348374, 6.94896516303808103610584257212, 7.89444003519395491309743448101, 8.61804711637913561356602154770, 9.21115137807534591060451794574, 10.16997336932811392977972722866, 10.730702154944031937314645146451, 11.54952707430541383475177732875, 12.36467142755024134303212988287, 13.46221505686136830476275166559, 14.2769897082442566423042149384, 14.75650847115899649941191248329, 15.93521325966005586940685944565, 16.613631769384699019113300074337, 17.372160783539193177100790610379, 17.57932331376033125192489173987, 18.58929834355888777756033431102, 19.28395724043717824736836458599, 20.159232730449574207071590523884

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