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Properties

Label 2-61-61.60-c7-0-15
Degree $2$
Conductor $61$
Sign $-0.990 - 0.140i$
Analytic cond. $19.0554$
Root an. cond. $4.36525$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 19.6i·2-s + 83.9·3-s − 256.·4-s + 90.0·5-s + 1.64e3i·6-s + 528. i·7-s − 2.51e3i·8-s + 4.86e3·9-s + 1.76e3i·10-s + 3.52e3i·11-s − 2.15e4·12-s − 8.20e3·13-s − 1.03e4·14-s + 7.56e3·15-s + 1.64e4·16-s + 1.56e4i·17-s + ⋯
L(s)  = 1  + 1.73i·2-s + 1.79·3-s − 2.00·4-s + 0.322·5-s + 3.11i·6-s + 0.582i·7-s − 1.73i·8-s + 2.22·9-s + 0.558i·10-s + 0.799i·11-s − 3.59·12-s − 1.03·13-s − 1.00·14-s + 0.578·15-s + 1.00·16-s + 0.771i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.140i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(61\)
Sign: $-0.990 - 0.140i$
Analytic conductor: \(19.0554\)
Root analytic conductor: \(4.36525\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (60, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 61,\ (\ :7/2),\ -0.990 - 0.140i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.220888 + 3.12477i\)
\(L(\frac12)\) \(\approx\) \(0.220888 + 3.12477i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad61 \( 1 + (1.75e6 + 2.49e5i)T \)
good2 \( 1 - 19.6iT - 128T^{2} \)
3 \( 1 - 83.9T + 2.18e3T^{2} \)
5 \( 1 - 90.0T + 7.81e4T^{2} \)
7 \( 1 - 528. iT - 8.23e5T^{2} \)
11 \( 1 - 3.52e3iT - 1.94e7T^{2} \)
13 \( 1 + 8.20e3T + 6.27e7T^{2} \)
17 \( 1 - 1.56e4iT - 4.10e8T^{2} \)
19 \( 1 - 3.41e4T + 8.93e8T^{2} \)
23 \( 1 - 8.07e4iT - 3.40e9T^{2} \)
29 \( 1 + 7.36e4iT - 1.72e10T^{2} \)
31 \( 1 + 1.37e5iT - 2.75e10T^{2} \)
37 \( 1 + 3.44e5iT - 9.49e10T^{2} \)
41 \( 1 - 8.43e5T + 1.94e11T^{2} \)
43 \( 1 + 4.51e5iT - 2.71e11T^{2} \)
47 \( 1 - 5.77e5T + 5.06e11T^{2} \)
53 \( 1 - 1.36e6iT - 1.17e12T^{2} \)
59 \( 1 + 2.44e6iT - 2.48e12T^{2} \)
67 \( 1 + 1.74e6iT - 6.06e12T^{2} \)
71 \( 1 - 3.90e5iT - 9.09e12T^{2} \)
73 \( 1 + 1.42e5T + 1.10e13T^{2} \)
79 \( 1 - 8.05e6iT - 1.92e13T^{2} \)
83 \( 1 - 2.78e6T + 2.71e13T^{2} \)
89 \( 1 - 6.03e6iT - 4.42e13T^{2} \)
97 \( 1 + 1.18e6T + 8.07e13T^{2} \)
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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

−14.30549389991135972237215754525, −13.68847231617638918236024207776, −12.52685840094146068498868599878, −9.605258888040769350101177902583, −9.268215314749380729829958789322, −7.83088125481747299450430523934, −7.38170330295807894089869130651, −5.63768988779197314129244622580, −4.08064971910960564542433401114, −2.22492507834116557104521578194, 0.994768720143387153584984167054, 2.45020766434063533287459873129, 3.23319235771636691806792595483, 4.53386058684117376296775555217, 7.48405097499536481343089526721, 8.817043893829924942241945401932, 9.645657661497301457197586894980, 10.48094559151559534856629342066, 11.99280475644396083981658725185, 13.16363803907602925187805477638

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