Properties

Label 2-240-5.4-c3-0-6
Degree 22
Conductor 240240
Sign 0.4470.894i0.447 - 0.894i
Analytic cond. 14.160414.1604
Root an. cond. 3.763033.76303
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)=(240s/2ΓC(s)L(s)=((0.4470.894i)Λ(4s)\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(240s/2ΓC(s+3/2)L(s)=((0.4470.894i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{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: 240240    =    24352^{4} \cdot 3 \cdot 5
Sign: 0.4470.894i0.447 - 0.894i
Analytic conductor: 14.160414.1604
Root analytic conductor: 3.763033.76303
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ240(49,)\chi_{240} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 240, ( :3/2), 0.4470.894i)(2,\ 240,\ (\ :3/2),\ 0.447 - 0.894i)

Particular Values

L(2)L(2) \approx 1.23325+0.762193i1.23325 + 0.762193i
L(12)L(\frac12) \approx 1.23325+0.762193i1.23325 + 0.762193i
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 13iT 1 - 3iT
5 1+(10+5i)T 1 + (10 + 5i)T
good7 1+10iT343T2 1 + 10iT - 343T^{2}
11 146T+1.33e3T2 1 - 46T + 1.33e3T^{2}
13 134iT2.19e3T2 1 - 34iT - 2.19e3T^{2}
17 166iT4.91e3T2 1 - 66iT - 4.91e3T^{2}
19 1104T+6.85e3T2 1 - 104T + 6.85e3T^{2}
23 1164iT1.21e4T2 1 - 164iT - 1.21e4T^{2}
29 1+224T+2.43e4T2 1 + 224T + 2.43e4T^{2}
31 172T+2.97e4T2 1 - 72T + 2.97e4T^{2}
37 1+22iT5.06e4T2 1 + 22iT - 5.06e4T^{2}
41 1194T+6.89e4T2 1 - 194T + 6.89e4T^{2}
43 1108iT7.95e4T2 1 - 108iT - 7.95e4T^{2}
47 1480iT1.03e5T2 1 - 480iT - 1.03e5T^{2}
53 1+286iT1.48e5T2 1 + 286iT - 1.48e5T^{2}
59 1426T+2.05e5T2 1 - 426T + 2.05e5T^{2}
61 1698T+2.26e5T2 1 - 698T + 2.26e5T^{2}
67 1+328iT3.00e5T2 1 + 328iT - 3.00e5T^{2}
71 1+188T+3.57e5T2 1 + 188T + 3.57e5T^{2}
73 1740iT3.89e5T2 1 - 740iT - 3.89e5T^{2}
79 11.16e3T+4.93e5T2 1 - 1.16e3T + 4.93e5T^{2}
83 1412iT5.71e5T2 1 - 412iT - 5.71e5T^{2}
89 1+1.20e3T+7.04e5T2 1 + 1.20e3T + 7.04e5T^{2}
97 1+1.38e3iT9.12e5T2 1 + 1.38e3iT - 9.12e5T^{2}
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   L(s)=p j=12(1αj,pps)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

−11.56443050906327530517220662032, −11.21397825538864847938256684229, −9.727043493866489091855245491148, −9.104298777559835004724024263880, −7.922391735247120713232863085489, −6.96437411932091214153296068620, −5.54211626932459469915570398594, −4.17196759682810013289975334958, −3.62648118495801271238709323632, −1.25387175509316289668548395249, 0.70148799396917029260058376361, 2.62346782158293543362061476002, 3.84986752814880570112904772219, 5.38159035300773826298257328281, 6.65502269083394473260254694649, 7.44222762645074758291071728648, 8.475669382080289593855659949869, 9.446368007447710611995930170548, 10.77716432500795205941830858081, 11.82743919665936893472680894648

Graph of the ZZ-function along the critical line