Properties

Label 2-61-61.52-c1-0-2
Degree $2$
Conductor $61$
Sign $0.999 + 0.0187i$
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  + (0.279 − 0.384i)2-s + (0.261 + 0.190i)3-s + (0.548 + 1.68i)4-s + (0.00765 + 0.0235i)5-s + (0.146 − 0.0475i)6-s + (−2.52 − 3.47i)7-s + (1.70 + 0.554i)8-s + (−0.894 − 2.75i)9-s + (0.0111 + 0.00363i)10-s + 5.23i·11-s + (−0.177 + 0.545i)12-s − 4.67·13-s − 2.04·14-s + (−0.00247 + 0.00762i)15-s + (−2.18 + 1.58i)16-s + (6.11 − 1.98i)17-s + ⋯
L(s)  = 1  + (0.197 − 0.271i)2-s + (0.151 + 0.109i)3-s + (0.274 + 0.843i)4-s + (0.00342 + 0.0105i)5-s + (0.0597 − 0.0194i)6-s + (−0.954 − 1.31i)7-s + (0.603 + 0.195i)8-s + (−0.298 − 0.917i)9-s + (0.00354 + 0.00115i)10-s + 1.57i·11-s + (−0.0512 + 0.157i)12-s − 1.29·13-s − 0.545·14-s + (−0.000639 + 0.00196i)15-s + (−0.545 + 0.396i)16-s + (1.48 − 0.481i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0187i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0187i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(61\)
Sign: $0.999 + 0.0187i$
Analytic conductor: \(0.487087\)
Root analytic conductor: \(0.697916\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (52, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 61,\ (\ :1/2),\ 0.999 + 0.0187i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.975762 - 0.00912798i\)
\(L(\frac12)\) \(\approx\) \(0.975762 - 0.00912798i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad61 \( 1 + (3.53 + 6.96i)T \)
good2 \( 1 + (-0.279 + 0.384i)T + (-0.618 - 1.90i)T^{2} \)
3 \( 1 + (-0.261 - 0.190i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (-0.00765 - 0.0235i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (2.52 + 3.47i)T + (-2.16 + 6.65i)T^{2} \)
11 \( 1 - 5.23iT - 11T^{2} \)
13 \( 1 + 4.67T + 13T^{2} \)
17 \( 1 + (-6.11 + 1.98i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.444 + 0.323i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (2.83 - 0.922i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 - 3.09iT - 29T^{2} \)
31 \( 1 + (-1.35 - 1.86i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.76 - 3.80i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-5.91 + 4.29i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + (3.17 + 1.03i)T + (34.7 + 25.2i)T^{2} \)
47 \( 1 - 1.49T + 47T^{2} \)
53 \( 1 + (-1.61 - 0.524i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (4.91 - 6.77i)T + (-18.2 - 56.1i)T^{2} \)
67 \( 1 + (5.02 - 1.63i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (8.01 + 2.60i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-2.62 + 8.08i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-8.24 - 2.67i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-3.02 - 2.20i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (2.24 - 3.09i)T + (-27.5 - 84.6i)T^{2} \)
97 \( 1 + (-0.159 + 0.115i)T + (29.9 - 92.2i)T^{2} \)
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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

−14.94771408458435037622760310218, −13.90024132816602151539920502944, −12.46834664661349595190525024730, −12.19382385623645805510435341210, −10.33424785459420140449305803699, −9.538479913582784930404315280271, −7.59218412362502445940752279320, −6.86425873055317073554284222225, −4.41627503291210573899992858802, −3.08246231793195354184388249713, 2.72406887281959213704636539487, 5.38643892840875761509012138385, 6.12716203106126679243216412650, 7.87855792819405271785193330688, 9.301714314031079072598511573722, 10.43149802482416774241279698981, 11.73735789668509476222410225353, 12.97045765361576498589678681121, 14.17600815485692434888160944748, 14.95719551257055633305698193406

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