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Properties

Label 1-19e2-361.196-r0-0-0
Degree $1$
Conductor $361$
Sign $-0.988 + 0.153i$
Analytic cond. $1.67647$
Root an. cond. $1.67647$
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  + (0.741 − 0.670i)2-s + (−0.0459 − 0.998i)3-s + (0.100 − 0.994i)4-s + (0.663 − 0.748i)5-s + (−0.703 − 0.710i)6-s + (−0.962 + 0.272i)7-s + (−0.592 − 0.805i)8-s + (−0.995 + 0.0917i)9-s + (−0.00918 − 0.999i)10-s + (0.904 + 0.426i)11-s + (−0.998 − 0.0550i)12-s + (−0.991 − 0.128i)13-s + (−0.531 + 0.847i)14-s + (−0.777 − 0.628i)15-s + (−0.979 − 0.200i)16-s + (−0.912 − 0.410i)17-s + ⋯
L(s)  = 1  + (0.741 − 0.670i)2-s + (−0.0459 − 0.998i)3-s + (0.100 − 0.994i)4-s + (0.663 − 0.748i)5-s + (−0.703 − 0.710i)6-s + (−0.962 + 0.272i)7-s + (−0.592 − 0.805i)8-s + (−0.995 + 0.0917i)9-s + (−0.00918 − 0.999i)10-s + (0.904 + 0.426i)11-s + (−0.998 − 0.0550i)12-s + (−0.991 − 0.128i)13-s + (−0.531 + 0.847i)14-s + (−0.777 − 0.628i)15-s + (−0.979 − 0.200i)16-s + (−0.912 − 0.410i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $-0.988 + 0.153i$
Analytic conductor: \(1.67647\)
Root analytic conductor: \(1.67647\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{361} (196, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 361,\ (0:\ ),\ -0.988 + 0.153i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1219830879 - 1.583710134i\)
\(L(\frac12)\) \(\approx\) \(-0.1219830879 - 1.583710134i\)
\(L(1)\) \(\approx\) \(0.7840719258 - 1.137861789i\)
\(L(1)\) \(\approx\) \(0.7840719258 - 1.137861789i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 + (0.741 - 0.670i)T \)
3 \( 1 + (-0.0459 - 0.998i)T \)
5 \( 1 + (0.663 - 0.748i)T \)
7 \( 1 + (-0.962 + 0.272i)T \)
11 \( 1 + (0.904 + 0.426i)T \)
13 \( 1 + (-0.991 - 0.128i)T \)
17 \( 1 + (-0.912 - 0.410i)T \)
23 \( 1 + (0.888 - 0.459i)T \)
29 \( 1 + (0.997 + 0.0734i)T \)
31 \( 1 + (-0.926 - 0.376i)T \)
37 \( 1 + (-0.0825 - 0.996i)T \)
41 \( 1 + (0.515 + 0.856i)T \)
43 \( 1 + (0.870 - 0.492i)T \)
47 \( 1 + (0.967 - 0.254i)T \)
53 \( 1 + (-0.263 + 0.964i)T \)
59 \( 1 + (0.484 - 0.875i)T \)
61 \( 1 + (0.0642 + 0.997i)T \)
67 \( 1 + (-0.971 - 0.236i)T \)
71 \( 1 + (0.0642 - 0.997i)T \)
73 \( 1 + (0.811 - 0.584i)T \)
79 \( 1 + (-0.861 - 0.507i)T \)
83 \( 1 + (0.0275 + 0.999i)T \)
89 \( 1 + (0.811 + 0.584i)T \)
97 \( 1 + (-0.971 + 0.236i)T \)
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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

−25.39549794001495077486307296855, −24.346265079339452664756074012112, −23.13160320091383204937551629780, −22.344313959589565617617531925669, −21.989389692647387078719466161180, −21.267023835134524774991499304977, −20.046548255537778075519591059446, −19.18546870420281459012105201719, −17.498440962121996380893157359867, −17.12242650331603436087144193421, −16.1537894053821664837367496518, −15.29491070862790358599431118748, −14.472616671999655536765165802235, −13.832416432533601566885312669226, −12.7725422261882806511884650950, −11.5642662875118127625033129955, −10.61711430342726002789794045651, −9.53395104244581943646591974314, −8.81075719195793760296097860714, −7.144883445092630762951471991904, −6.427213467487091058905662195518, −5.55026530381249737985617843376, −4.36110198501017847293763395424, −3.41024699340055431212137124958, −2.57281641263392005561115277945, 0.735145139009959125118766684437, 2.047843265283013661805868282776, 2.79395649118206596984148526128, 4.36167655884802246992493194406, 5.45098219206534097650573923531, 6.35981757425172226038090640086, 7.13225579649380035027330356357, 8.999543035652409003270681781638, 9.44063020137738762552779367939, 10.75966293225749135744653478609, 12.077797225076887466637003873351, 12.48510502931956683077902498858, 13.227229520516074800003050361331, 14.04384569649907218928552819282, 14.97846666319406177221578370852, 16.30078178386845616123873682116, 17.302879956116052128527088518370, 18.18486523169485980590821752437, 19.3491183743714645351422680050, 19.79431815326074164127625237848, 20.5886310156783893047062727905, 21.89283522416721449057141739447, 22.41285193696259135116227312852, 23.28461583549223669753691965432, 24.35818772200342171743638174250

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