Properties

Label 2-61-61.3-c1-0-1
Degree $2$
Conductor $61$
Sign $0.391 - 0.920i$
Analytic cond. $0.487087$
Root an. cond. $0.697916$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.186 − 0.0606i)2-s + (−0.865 + 2.66i)3-s + (−1.58 − 1.15i)4-s + (2.93 + 2.13i)5-s + (0.322 − 0.444i)6-s + (−0.222 + 0.0723i)7-s + (0.456 + 0.628i)8-s + (−3.91 − 2.84i)9-s + (−0.417 − 0.575i)10-s − 5.67i·11-s + (4.44 − 3.22i)12-s + 2.04·13-s + 0.0459·14-s + (−8.20 + 5.96i)15-s + (1.16 + 3.58i)16-s + (1.75 − 2.41i)17-s + ⋯
L(s)  = 1  + (−0.131 − 0.0428i)2-s + (−0.499 + 1.53i)3-s + (−0.793 − 0.576i)4-s + (1.31 + 0.952i)5-s + (0.131 − 0.181i)6-s + (−0.0841 + 0.0273i)7-s + (0.161 + 0.222i)8-s + (−1.30 − 0.947i)9-s + (−0.132 − 0.181i)10-s − 1.71i·11-s + (1.28 − 0.931i)12-s + 0.565·13-s + 0.0122·14-s + (−2.11 + 1.53i)15-s + (0.291 + 0.896i)16-s + (0.425 − 0.585i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.391 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(61\)
Sign: $0.391 - 0.920i$
Analytic conductor: \(0.487087\)
Root analytic conductor: \(0.697916\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 61,\ (\ :1/2),\ 0.391 - 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.616255 + 0.407747i\)
\(L(\frac12)\) \(\approx\) \(0.616255 + 0.407747i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad61 \( 1 + (7.54 - 2.01i)T \)
good2 \( 1 + (0.186 + 0.0606i)T + (1.61 + 1.17i)T^{2} \)
3 \( 1 + (0.865 - 2.66i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-2.93 - 2.13i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.222 - 0.0723i)T + (5.66 - 4.11i)T^{2} \)
11 \( 1 + 5.67iT - 11T^{2} \)
13 \( 1 - 2.04T + 13T^{2} \)
17 \( 1 + (-1.75 + 2.41i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.260 - 0.801i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (2.06 - 2.84i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + 3.91iT - 29T^{2} \)
31 \( 1 + (3.42 - 1.11i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (1.13 - 0.368i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.18 + 3.65i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (5.62 + 7.74i)T + (-13.2 + 40.8i)T^{2} \)
47 \( 1 - 6.21T + 47T^{2} \)
53 \( 1 + (1.35 + 1.86i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-4.75 - 1.54i)T + (47.7 + 34.6i)T^{2} \)
67 \( 1 + (1.99 - 2.74i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (1.12 + 1.55i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (4.14 - 3.01i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-6.05 - 8.33i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (3.02 - 9.32i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-6.67 - 2.16i)T + (72.0 + 52.3i)T^{2} \)
97 \( 1 + (-3.58 - 11.0i)T + (-78.4 + 57.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.23973070537076784224102490824, −14.09363171315168027701664179721, −13.64947038541836528637633571709, −11.25623578539732775881486783918, −10.48984340512639276876068221685, −9.761736100951160921822227907588, −8.820453785011287534860333014533, −6.02859172794265792170980537889, −5.43166616976039355140778770504, −3.55115321486051542253146341599, 1.68183952835725434680805745947, 4.90488961792261668237958826854, 6.28601828688923696022051831601, 7.63788518839913418535350083823, 8.867744056598731147559522924949, 10.02987113400393642598632011248, 12.10905680943220180176899470065, 12.86319770983173924433676212848, 13.21426826370634321047100860981, 14.39205846864938031524952053712

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