Properties

Label 1-19e2-361.178-r0-0-0
Degree $1$
Conductor $361$
Sign $0.999 + 0.0174i$
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  + (0.451 + 0.892i)2-s + (−0.754 + 0.656i)3-s + (−0.592 + 0.805i)4-s + (−0.298 − 0.954i)5-s + (−0.926 − 0.376i)6-s + (−0.401 + 0.915i)7-s + (−0.986 − 0.164i)8-s + (0.137 − 0.990i)9-s + (0.716 − 0.697i)10-s + (0.789 − 0.614i)11-s + (−0.0825 − 0.996i)12-s + (−0.191 − 0.981i)13-s + (−0.998 + 0.0550i)14-s + (0.851 + 0.523i)15-s + (−0.298 − 0.954i)16-s + (−0.592 − 0.805i)17-s + ⋯
L(s)  = 1  + (0.451 + 0.892i)2-s + (−0.754 + 0.656i)3-s + (−0.592 + 0.805i)4-s + (−0.298 − 0.954i)5-s + (−0.926 − 0.376i)6-s + (−0.401 + 0.915i)7-s + (−0.986 − 0.164i)8-s + (0.137 − 0.990i)9-s + (0.716 − 0.697i)10-s + (0.789 − 0.614i)11-s + (−0.0825 − 0.996i)12-s + (−0.191 − 0.981i)13-s + (−0.998 + 0.0550i)14-s + (0.851 + 0.523i)15-s + (−0.298 − 0.954i)16-s + (−0.592 − 0.805i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0174i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0174i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $0.999 + 0.0174i$
Analytic conductor: \(1.67647\)
Root analytic conductor: \(1.67647\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{361} (178, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 361,\ (0:\ ),\ 0.999 + 0.0174i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7548057825 + 0.006568842561i\)
\(L(\frac12)\) \(\approx\) \(0.7548057825 + 0.006568842561i\)
\(L(1)\) \(\approx\) \(0.7541695321 + 0.3060240301i\)
\(L(1)\) \(\approx\) \(0.7541695321 + 0.3060240301i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 + (0.451 + 0.892i)T \)
3 \( 1 + (-0.754 + 0.656i)T \)
5 \( 1 + (-0.298 - 0.954i)T \)
7 \( 1 + (-0.401 + 0.915i)T \)
11 \( 1 + (0.789 - 0.614i)T \)
13 \( 1 + (-0.191 - 0.981i)T \)
17 \( 1 + (-0.592 - 0.805i)T \)
23 \( 1 + (-0.754 - 0.656i)T \)
29 \( 1 + (0.993 - 0.110i)T \)
31 \( 1 + (0.546 + 0.837i)T \)
37 \( 1 + (0.789 - 0.614i)T \)
41 \( 1 + (0.0275 - 0.999i)T \)
43 \( 1 + (0.716 + 0.697i)T \)
47 \( 1 + (-0.926 - 0.376i)T \)
53 \( 1 + (-0.926 - 0.376i)T \)
59 \( 1 + (0.0275 - 0.999i)T \)
61 \( 1 + (0.635 + 0.771i)T \)
67 \( 1 + (0.350 + 0.936i)T \)
71 \( 1 + (0.635 - 0.771i)T \)
73 \( 1 + (-0.592 - 0.805i)T \)
79 \( 1 + (0.716 + 0.697i)T \)
83 \( 1 + (-0.677 - 0.735i)T \)
89 \( 1 + (-0.592 + 0.805i)T \)
97 \( 1 + (0.350 - 0.936i)T \)
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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

−24.228852716723220860836833051865, −23.57752694673046581265362638313, −22.90820407669135952010176977628, −22.20349653094713427580006885409, −21.55407185476993460181120205340, −19.99303654329797608632002119997, −19.5085253991706025259642644530, −18.7675719507872489423248450007, −17.75600682662004440253154193893, −17.06481663768779047187427040661, −15.712912362917150156354438314588, −14.48436846652046174241315765818, −13.81369792536122892413900402429, −12.89565412795940104285846109915, −11.855366986541932040153780624600, −11.31088841452567384669611703008, −10.37224238872893743117172140387, −9.602888266825024252342698711106, −7.872288143333547969044382303781, −6.62996608233827185887980360637, −6.29356224452998929756938094644, −4.551817622934487141520835241944, −3.875445580750280617887602572591, −2.42058379828958691702710755011, −1.324673323064686990343990377279, 0.47847165431423186647253561922, 3.022629965784997050426638227, 4.18030371932438488388416648362, 5.05262008558042591460539841334, 5.835853268851873841383053287365, 6.66689897123186018071274763222, 8.21547707572880626628714237194, 8.953150940710100119772504915070, 9.79513718308475556283549595914, 11.41481616443894475582861814152, 12.22605766348629470416643782403, 12.80357122522622779237468337145, 14.11469209611008663820251870376, 15.20907852789130030095008197280, 16.031434190849045510254598683095, 16.28482237417490444055973341984, 17.45826175003630871070453994187, 18.04166282637053843929342919843, 19.47610762944391452946206566685, 20.63070918242510149672592441701, 21.53003451893863971842560870056, 22.297832144960560380198115552009, 22.84509500321425467642813431643, 23.89979108688419939719927140618, 24.73478231930285056980486554627

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