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Properties

Label 2-61-61.60-c3-0-8
Degree $2$
Conductor $61$
Sign $0.902 - 0.431i$
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  + 1.28i·2-s + 5.54·3-s + 6.33·4-s + 0.0273·5-s + 7.15i·6-s − 21.0i·7-s + 18.4i·8-s + 3.77·9-s + 0.0352i·10-s + 20.2i·11-s + 35.1·12-s − 9.47·13-s + 27.0·14-s + 0.151·15-s + 26.8·16-s + 90.4i·17-s + ⋯
L(s)  = 1  + 0.455i·2-s + 1.06·3-s + 0.792·4-s + 0.00244·5-s + 0.486i·6-s − 1.13i·7-s + 0.816i·8-s + 0.139·9-s + 0.00111i·10-s + 0.554i·11-s + 0.845·12-s − 0.202·13-s + 0.516·14-s + 0.00260·15-s + 0.419·16-s + 1.28i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 - 0.431i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.902 - 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(61\)
Sign: $0.902 - 0.431i$
Analytic conductor: \(3.59911\)
Root analytic conductor: \(1.89713\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (60, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 61,\ (\ :3/2),\ 0.902 - 0.431i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.09580 + 0.474967i\)
\(L(\frac12)\) \(\approx\) \(2.09580 + 0.474967i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad61 \( 1 + (-429. + 205. i)T \)
good2 \( 1 - 1.28iT - 8T^{2} \)
3 \( 1 - 5.54T + 27T^{2} \)
5 \( 1 - 0.0273T + 125T^{2} \)
7 \( 1 + 21.0iT - 343T^{2} \)
11 \( 1 - 20.2iT - 1.33e3T^{2} \)
13 \( 1 + 9.47T + 2.19e3T^{2} \)
17 \( 1 - 90.4iT - 4.91e3T^{2} \)
19 \( 1 + 54.9T + 6.85e3T^{2} \)
23 \( 1 + 142. iT - 1.21e4T^{2} \)
29 \( 1 + 115. iT - 2.43e4T^{2} \)
31 \( 1 + 104. iT - 2.97e4T^{2} \)
37 \( 1 - 73.2iT - 5.06e4T^{2} \)
41 \( 1 - 206.T + 6.89e4T^{2} \)
43 \( 1 - 17.9iT - 7.95e4T^{2} \)
47 \( 1 - 36.1T + 1.03e5T^{2} \)
53 \( 1 - 361. iT - 1.48e5T^{2} \)
59 \( 1 + 78.4iT - 2.05e5T^{2} \)
67 \( 1 + 1.03e3iT - 3.00e5T^{2} \)
71 \( 1 - 837. iT - 3.57e5T^{2} \)
73 \( 1 + 13.7T + 3.89e5T^{2} \)
79 \( 1 - 1.10e3iT - 4.93e5T^{2} \)
83 \( 1 - 503.T + 5.71e5T^{2} \)
89 \( 1 + 308. iT - 7.04e5T^{2} \)
97 \( 1 + 1.43e3T + 9.12e5T^{2} \)
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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

−14.72142497803033938459256458268, −13.84060545734775196464853348170, −12.58894101233325111020693860345, −11.05417045485061312468461758669, −9.989357629563860442393156734231, −8.351593638714308159554631563093, −7.54134203408270348150228077313, −6.25692608024899616837936842375, −4.03949345149728108010848710924, −2.24090335577363794915223593313, 2.21616881917966661928629010724, 3.29803230672190927781028317624, 5.72950714208638973360461705047, 7.38292162440022704272379263391, 8.698529377890811426794019651559, 9.675246585745161552969872584570, 11.24927373519695222472438829539, 12.07690905643321600566612378457, 13.36974760308016454693979031289, 14.52990767545197805362162454412

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