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Properties

Label 1-19e2-361.11-r0-0-0
Degree $1$
Conductor $361$
Sign $-0.780 - 0.625i$
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  + (−0.298 + 0.954i)2-s + (−0.962 + 0.272i)3-s + (−0.821 − 0.569i)4-s + (0.350 + 0.936i)5-s + (0.0275 − 0.999i)6-s + (−0.0825 − 0.996i)7-s + (0.789 − 0.614i)8-s + (0.851 − 0.523i)9-s + (−0.998 + 0.0550i)10-s + (−0.879 + 0.475i)11-s + (0.945 + 0.324i)12-s + (0.716 + 0.697i)13-s + (0.975 + 0.218i)14-s + (−0.592 − 0.805i)15-s + (0.350 + 0.936i)16-s + (−0.821 + 0.569i)17-s + ⋯
L(s)  = 1  + (−0.298 + 0.954i)2-s + (−0.962 + 0.272i)3-s + (−0.821 − 0.569i)4-s + (0.350 + 0.936i)5-s + (0.0275 − 0.999i)6-s + (−0.0825 − 0.996i)7-s + (0.789 − 0.614i)8-s + (0.851 − 0.523i)9-s + (−0.998 + 0.0550i)10-s + (−0.879 + 0.475i)11-s + (0.945 + 0.324i)12-s + (0.716 + 0.697i)13-s + (0.975 + 0.218i)14-s + (−0.592 − 0.805i)15-s + (0.350 + 0.936i)16-s + (−0.821 + 0.569i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.780 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.780 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $-0.780 - 0.625i$
Analytic conductor: \(1.67647\)
Root analytic conductor: \(1.67647\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{361} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 361,\ (0:\ ),\ -0.780 - 0.625i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1038467300 + 0.2954996731i\)
\(L(\frac12)\) \(\approx\) \(-0.1038467300 + 0.2954996731i\)
\(L(1)\) \(\approx\) \(0.4058972094 + 0.3466680757i\)
\(L(1)\) \(\approx\) \(0.4058972094 + 0.3466680757i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 + (-0.298 + 0.954i)T \)
3 \( 1 + (-0.962 + 0.272i)T \)
5 \( 1 + (0.350 + 0.936i)T \)
7 \( 1 + (-0.0825 - 0.996i)T \)
11 \( 1 + (-0.879 + 0.475i)T \)
13 \( 1 + (0.716 + 0.697i)T \)
17 \( 1 + (-0.821 + 0.569i)T \)
23 \( 1 + (-0.962 - 0.272i)T \)
29 \( 1 + (0.904 + 0.426i)T \)
31 \( 1 + (-0.677 + 0.735i)T \)
37 \( 1 + (-0.879 + 0.475i)T \)
41 \( 1 + (0.993 - 0.110i)T \)
43 \( 1 + (-0.998 - 0.0550i)T \)
47 \( 1 + (0.0275 - 0.999i)T \)
53 \( 1 + (0.0275 - 0.999i)T \)
59 \( 1 + (0.993 - 0.110i)T \)
61 \( 1 + (-0.926 + 0.376i)T \)
67 \( 1 + (0.137 + 0.990i)T \)
71 \( 1 + (-0.926 - 0.376i)T \)
73 \( 1 + (-0.821 + 0.569i)T \)
79 \( 1 + (-0.998 - 0.0550i)T \)
83 \( 1 + (-0.986 + 0.164i)T \)
89 \( 1 + (-0.821 - 0.569i)T \)
97 \( 1 + (0.137 - 0.990i)T \)
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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

−24.13075811589215048153071606306, −23.14977862812686880048594539410, −22.241531531614870917067989603833, −21.500897612282234781458232908598, −20.82589880633499618748703598627, −19.80419191505354994269178766252, −18.69036118559179272960422886868, −18.033749728559583850627634629856, −17.47387346972971191631919488891, −16.21379930910572048771135605415, −15.723007673504029350516824124914, −13.70576639129287923385352440606, −13.0503162284092275644706792867, −12.34338929179064306318197333266, −11.520075457405851279981988867612, −10.63296975930229207485424667487, −9.62988437414150830487967831308, −8.63896376890134286278672210101, −7.80648568756216085180380157717, −5.98792875960168766774470552070, −5.34251288356053417972937915336, −4.32988178622495637659035037611, −2.69845092710536571011977041125, −1.60867915762428925535800861272, −0.24504308583620182816263085561, 1.630842070638409900528056082093, 3.75387622923743889417547302362, 4.67457976436010487569418424384, 5.84473570806536248378973249686, 6.72758960066914188601022760650, 7.21838487727827646009640936111, 8.60744983614228359291642599828, 10.1232693411232621391164179809, 10.314019435389249075236431462374, 11.31303221921795737514681450062, 12.89626947554696153613066144529, 13.731394916971411436557509929624, 14.66243848804224032462542703019, 15.72166532635252348846574518549, 16.29876023442081103471902319555, 17.37924846936098585024982808430, 17.933852422052555925594712786830, 18.59440664163004812683513912535, 19.78456345608939194212022320108, 21.15777413504810834293834807296, 22.05788559951082497944284437390, 22.87052564516904441219675788548, 23.53434315376093989870242072071, 24.05095230428140069036569894851, 25.49006774099932617742580618463

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