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Properties

Label 2-960-5.4-c3-0-54
Degree 22
Conductor 960960
Sign 0.447+0.894i0.447 + 0.894i
Analytic cond. 56.641856.6418
Root an. cond. 7.526077.52607
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3i·3-s + (10 − 5i)5-s + 10i·7-s − 9·9-s − 46·11-s + 34i·13-s + (15 + 30i)15-s − 66i·17-s − 104·19-s − 30·21-s − 164i·23-s + (75 − 100i)25-s − 27i·27-s + 224·29-s + 72·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.894 − 0.447i)5-s + 0.539i·7-s − 0.333·9-s − 1.26·11-s + 0.725i·13-s + (0.258 + 0.516i)15-s − 0.941i·17-s − 1.25·19-s − 0.311·21-s − 1.48i·23-s + (0.599 − 0.800i)25-s − 0.192i·27-s + 1.43·29-s + 0.417·31-s + ⋯

Functional equation

Λ(s)=(960s/2 ΓC(s) L(s)=((0.447+0.894i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(960s/2 ΓC(s+3/2) L(s)=((0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 960960    =    26⋅3⋅52^{6} \cdot 3 \cdot 5
Sign: 0.447+0.894i0.447 + 0.894i
Analytic conductor: 56.641856.6418
Root analytic conductor: 7.526077.52607
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ960(769,⋅)\chi_{960} (769, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 960, ( :3/2), 0.447+0.894i)(2,\ 960,\ (\ :3/2),\ 0.447 + 0.894i)

Particular Values

L(2)L(2) ≈\approx 1.5377221381.537722138
L(12)L(\frac12) ≈\approx 1.5377221381.537722138
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1−3iT 1 - 3iT
5 1+(−10+5i)T 1 + (-10 + 5i)T
good7 1−10iT−343T2 1 - 10iT - 343T^{2}
11 1+46T+1.33e3T2 1 + 46T + 1.33e3T^{2}
13 1−34iT−2.19e3T2 1 - 34iT - 2.19e3T^{2}
17 1+66iT−4.91e3T2 1 + 66iT - 4.91e3T^{2}
19 1+104T+6.85e3T2 1 + 104T + 6.85e3T^{2}
23 1+164iT−1.21e4T2 1 + 164iT - 1.21e4T^{2}
29 1−224T+2.43e4T2 1 - 224T + 2.43e4T^{2}
31 1−72T+2.97e4T2 1 - 72T + 2.97e4T^{2}
37 1+22iT−5.06e4T2 1 + 22iT - 5.06e4T^{2}
41 1−194T+6.89e4T2 1 - 194T + 6.89e4T^{2}
43 1−108iT−7.95e4T2 1 - 108iT - 7.95e4T^{2}
47 1+480iT−1.03e5T2 1 + 480iT - 1.03e5T^{2}
53 1+286iT−1.48e5T2 1 + 286iT - 1.48e5T^{2}
59 1+426T+2.05e5T2 1 + 426T + 2.05e5T^{2}
61 1+698T+2.26e5T2 1 + 698T + 2.26e5T^{2}
67 1+328iT−3.00e5T2 1 + 328iT - 3.00e5T^{2}
71 1+188T+3.57e5T2 1 + 188T + 3.57e5T^{2}
73 1+740iT−3.89e5T2 1 + 740iT - 3.89e5T^{2}
79 1−1.16e3T+4.93e5T2 1 - 1.16e3T + 4.93e5T^{2}
83 1−412iT−5.71e5T2 1 - 412iT - 5.71e5T^{2}
89 1+1.20e3T+7.04e5T2 1 + 1.20e3T + 7.04e5T^{2}
97 1−1.38e3iT−9.12e5T2 1 - 1.38e3iT - 9.12e5T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−9.467437670454232112234383501244, −8.766644010844473602937549864723, −8.126768815843957617172444397341, −6.72185089074116046775563195391, −5.98711442494287936637141828110, −4.95904737334328059198075661642, −4.49983328596882906459509705342, −2.80072094046420710867696922456, −2.15704981035420949541059276785, −0.39508833229165773712208172791, 1.14301097533282647415889360878, 2.30957186702099164991215068515, 3.17605593430784549350289147412, 4.59009741789531277580639603400, 5.73895342183115544031714318363, 6.24791699268170186524000313511, 7.34321019391167641477339616812, 7.958821920559701306408864216731, 8.883251106493685934255539400981, 10.05718288977837683351891995446

Graph of the ZZ-function along the critical line