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Properties

Label 2-61-61.27-c1-0-2
Degree $2$
Conductor $61$
Sign $0.214 + 0.976i$
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  + (−0.737 − 1.01i)2-s + (−0.446 + 0.324i)3-s + (0.131 − 0.403i)4-s + (0.620 − 1.91i)5-s + (0.659 + 0.214i)6-s + (1.40 − 1.93i)7-s + (−2.89 + 0.940i)8-s + (−0.832 + 2.56i)9-s + (−2.39 + 0.779i)10-s + 3.52i·11-s + (0.0724 + 0.222i)12-s + 4.88·13-s − 3.00·14-s + (0.342 + 1.05i)15-s + (2.40 + 1.74i)16-s + (−0.242 − 0.0788i)17-s + ⋯
L(s)  = 1  + (−0.521 − 0.718i)2-s + (−0.257 + 0.187i)3-s + (0.0655 − 0.201i)4-s + (0.277 − 0.854i)5-s + (0.269 + 0.0874i)6-s + (0.531 − 0.730i)7-s + (−1.02 + 0.332i)8-s + (−0.277 + 0.854i)9-s + (−0.758 + 0.246i)10-s + 1.06i·11-s + (0.0209 + 0.0643i)12-s + 1.35·13-s − 0.801·14-s + (0.0884 + 0.272i)15-s + (0.600 + 0.436i)16-s + (−0.0588 − 0.0191i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.214 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.214 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(61\)
Sign: $0.214 + 0.976i$
Analytic conductor: \(0.487087\)
Root analytic conductor: \(0.697916\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 61,\ (\ :1/2),\ 0.214 + 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.543405 - 0.437066i\)
\(L(\frac12)\) \(\approx\) \(0.543405 - 0.437066i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad61 \( 1 + (7.52 + 2.09i)T \)
good2 \( 1 + (0.737 + 1.01i)T + (-0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.446 - 0.324i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (-0.620 + 1.91i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-1.40 + 1.93i)T + (-2.16 - 6.65i)T^{2} \)
11 \( 1 - 3.52iT - 11T^{2} \)
13 \( 1 - 4.88T + 13T^{2} \)
17 \( 1 + (0.242 + 0.0788i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (4.30 - 3.12i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (1.09 + 0.356i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + 6.09iT - 29T^{2} \)
31 \( 1 + (2.59 - 3.57i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (-5.80 + 7.98i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-0.148 - 0.107i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (5.29 - 1.72i)T + (34.7 - 25.2i)T^{2} \)
47 \( 1 - 3.83T + 47T^{2} \)
53 \( 1 + (13.6 - 4.42i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-3.02 - 4.16i)T + (-18.2 + 56.1i)T^{2} \)
67 \( 1 + (11.9 + 3.87i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (-1.81 + 0.589i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (0.832 + 2.56i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (7.35 - 2.39i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (-12.5 + 9.10i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-9.09 - 12.5i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (-3.36 - 2.44i)T + (29.9 + 92.2i)T^{2} \)
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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

−14.80321546423157862713591742127, −13.58885888190203087504041943502, −12.40742953753499415284490622679, −11.07089514319195238814296035191, −10.45733342135350426515894403757, −9.195474809625363348980033230212, −7.994091022216452060025859236100, −5.95302147281430531350262708447, −4.49542464750220258627738104194, −1.70640426211790740564416212753, 3.20495135061973258084180686002, 5.98851310714846896092967300189, 6.62666146775149152171880500042, 8.305069376671871768476970270180, 9.040606602902004485067523381053, 10.91910753195443207205354577876, 11.73925471841922690442049669556, 13.12779552558778892893176274598, 14.57372651018457283623546974608, 15.34172474975214786915457292725

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