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Properties

Label 2-61-61.48-c3-0-4
Degree $2$
Conductor $61$
Sign $0.475 + 0.879i$
Analytic cond. $3.59911$
Root an. cond. $1.89713$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.43 − 2.56i)2-s − 4.68·3-s + (9.13 + 15.8i)4-s + (−2.08 + 3.60i)5-s + (20.8 + 12.0i)6-s + (7.86 + 4.54i)7-s − 52.5i·8-s − 5.02·9-s + (18.4 − 10.6i)10-s − 18.5i·11-s + (−42.8 − 74.1i)12-s + (19.4 − 33.7i)13-s + (−23.2 − 40.3i)14-s + (9.76 − 16.9i)15-s + (−61.7 + 106. i)16-s + (87.3 − 50.4i)17-s + ⋯
L(s)  = 1  + (−1.56 − 0.905i)2-s − 0.902·3-s + (1.14 + 1.97i)4-s + (−0.186 + 0.322i)5-s + (1.41 + 0.817i)6-s + (0.424 + 0.245i)7-s − 2.32i·8-s − 0.186·9-s + (0.584 − 0.337i)10-s − 0.507i·11-s + (−1.02 − 1.78i)12-s + (0.415 − 0.719i)13-s + (−0.444 − 0.769i)14-s + (0.168 − 0.291i)15-s + (−0.964 + 1.67i)16-s + (1.24 − 0.719i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.475 + 0.879i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.475 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(61\)
Sign: $0.475 + 0.879i$
Analytic conductor: \(3.59911\)
Root analytic conductor: \(1.89713\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (48, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 61,\ (\ :3/2),\ 0.475 + 0.879i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.407795 - 0.243275i\)
\(L(\frac12)\) \(\approx\) \(0.407795 - 0.243275i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad61 \( 1 + (266. - 394. i)T \)
good2 \( 1 + (4.43 + 2.56i)T + (4 + 6.92i)T^{2} \)
3 \( 1 + 4.68T + 27T^{2} \)
5 \( 1 + (2.08 - 3.60i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (-7.86 - 4.54i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + 18.5iT - 1.33e3T^{2} \)
13 \( 1 + (-19.4 + 33.7i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-87.3 + 50.4i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-65.2 - 113. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + 139. iT - 1.21e4T^{2} \)
29 \( 1 + (54.4 - 31.4i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-178. + 102. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 200. iT - 5.06e4T^{2} \)
41 \( 1 - 98.4T + 6.89e4T^{2} \)
43 \( 1 + (26.6 + 15.3i)T + (3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-296. - 514. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 318. iT - 1.48e5T^{2} \)
59 \( 1 + (-181. - 104. i)T + (1.02e5 + 1.77e5i)T^{2} \)
67 \( 1 + (-626. - 361. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (528. - 304. i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (478. + 829. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-580. - 335. i)T + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (26.7 - 46.3i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 1.02e3iT - 7.04e5T^{2} \)
97 \( 1 + (470. + 814. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−14.41963515012551271449588647538, −12.47362992876978819380796648827, −11.66122752586964666200667453665, −10.89682203550201375175116555433, −9.988232175077810111454089725817, −8.552223966406634097623171027260, −7.55708235763401986120833830796, −5.75213421743148643263280634994, −3.07920726113584125678761443271, −0.845404468002001370612978631738, 1.05443998656541313753353851158, 5.14386214948407679356290494368, 6.42514927987489029561513358968, 7.60403856096345590753175053967, 8.747910720040841743512625771770, 9.941010648196742253457890508363, 11.05918699227162130210983075543, 11.97654687696489614275140151344, 13.99414117197614406274402070081, 15.27859497011435443243468865471

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