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Properties

Label 1-61-61.51-r1-0-0
Degree $1$
Conductor $61$
Sign $-0.381 + 0.924i$
Analytic cond. $6.55536$
Root an. cond. $6.55536$
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  + (0.743 − 0.669i)2-s + (−0.309 + 0.951i)3-s + (0.104 − 0.994i)4-s + (−0.913 + 0.406i)5-s + (0.406 + 0.913i)6-s + (−0.207 + 0.978i)7-s + (−0.587 − 0.809i)8-s + (−0.809 − 0.587i)9-s + (−0.406 + 0.913i)10-s + i·11-s + (0.913 + 0.406i)12-s + (−0.5 + 0.866i)13-s + (0.5 + 0.866i)14-s + (−0.104 − 0.994i)15-s + (−0.978 − 0.207i)16-s + (−0.994 − 0.104i)17-s + ⋯
L(s)  = 1  + (0.743 − 0.669i)2-s + (−0.309 + 0.951i)3-s + (0.104 − 0.994i)4-s + (−0.913 + 0.406i)5-s + (0.406 + 0.913i)6-s + (−0.207 + 0.978i)7-s + (−0.587 − 0.809i)8-s + (−0.809 − 0.587i)9-s + (−0.406 + 0.913i)10-s + i·11-s + (0.913 + 0.406i)12-s + (−0.5 + 0.866i)13-s + (0.5 + 0.866i)14-s + (−0.104 − 0.994i)15-s + (−0.978 − 0.207i)16-s + (−0.994 − 0.104i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.381 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.381 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(61\)
Sign: $-0.381 + 0.924i$
Analytic conductor: \(6.55536\)
Root analytic conductor: \(6.55536\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 61,\ (1:\ ),\ -0.381 + 0.924i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5622527194 + 0.8398844043i\)
\(L(\frac12)\) \(\approx\) \(0.5622527194 + 0.8398844043i\)
\(L(1)\) \(\approx\) \(0.9610200873 + 0.2106333001i\)
\(L(1)\) \(\approx\) \(0.9610200873 + 0.2106333001i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad61 \( 1 \)
good2 \( 1 + (0.743 - 0.669i)T \)
3 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + (-0.913 + 0.406i)T \)
7 \( 1 + (-0.207 + 0.978i)T \)
11 \( 1 + iT \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (-0.994 - 0.104i)T \)
19 \( 1 + (0.978 - 0.207i)T \)
23 \( 1 + (-0.587 + 0.809i)T \)
29 \( 1 + (0.866 - 0.5i)T \)
31 \( 1 + (0.743 + 0.669i)T \)
37 \( 1 + (-0.951 + 0.309i)T \)
41 \( 1 + (-0.309 - 0.951i)T \)
43 \( 1 + (0.994 - 0.104i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (0.587 + 0.809i)T \)
59 \( 1 + (-0.743 + 0.669i)T \)
67 \( 1 + (0.406 + 0.913i)T \)
71 \( 1 + (-0.406 + 0.913i)T \)
73 \( 1 + (0.913 + 0.406i)T \)
79 \( 1 + (-0.994 + 0.104i)T \)
83 \( 1 + (0.669 + 0.743i)T \)
89 \( 1 + (0.951 + 0.309i)T \)
97 \( 1 + (-0.669 + 0.743i)T \)
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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

−31.90451400205936337969736109611, −30.86077990666884292530533544543, −29.95892016609046196959177821312, −28.96696978338933579393680869771, −27.20487522107507963590011332379, −26.24342538894359015332134105167, −24.534654744132605957210464524262, −24.23869236805388959227592205354, −23.05742890866795567745372702097, −22.33781010501028387309185118209, −20.42250601064951026701762860978, −19.51835460688858440682160635148, −17.86754852422768716396651879476, −16.742795145918319698742602471749, −15.84126415823026252126133157668, −14.184953046212135111116082363825, −13.200776302324380229576414938630, −12.194530009379584256679800896376, −11.02679561165005507085906962273, −8.356447452305678679788218025334, −7.52709652781601669448169132043, −6.31841569869582289993012512434, −4.77900077532981463696358703468, −3.19685551704739142963365697285, −0.428900700201123469156532859520, 2.60140474976288679169136542055, 4.04484181253022934237017860826, 5.11668827153458069628401132432, 6.742425335434052280544222932196, 9.08608201003009770150767193013, 10.22154821896370512241416627725, 11.67414910245414845497334953923, 12.091156593546437967869032470720, 14.12438137181192367739037270329, 15.3969876806554631423255904507, 15.75286840348756507102048927482, 17.87031793010275540422173280991, 19.30202579395962108254344668022, 20.23276557953752334619408740214, 21.56280443397120097691018510958, 22.34903755971789706330811277387, 23.1544276394424461109080970191, 24.45447428781395356842367167738, 26.16169098733232618212205368631, 27.38065821265833932748885358127, 28.29330579064070724204040047183, 29.064753513432059009485928775503, 30.89476039131371939076183225550, 31.29713275111278603370884051596, 32.40996882654193317823354217361

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