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Properties

Label 2-3240-9.7-c1-0-24
Degree 22
Conductor 32403240
Sign 0.766+0.642i0.766 + 0.642i
Analytic cond. 25.871525.8715
Root an. cond. 5.086405.08640
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)5-s + (−1 + 1.73i)7-s + (1 − 1.73i)11-s + (−2 − 3.46i)13-s + 2·17-s + 4·19-s + (4 + 6.92i)23-s + (−0.499 + 0.866i)25-s + (−5 + 8.66i)29-s + (−2 − 3.46i)31-s + 1.99·35-s + (4 − 6.92i)43-s + (4 − 6.92i)47-s + (1.50 + 2.59i)49-s − 6·53-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (−0.377 + 0.654i)7-s + (0.301 − 0.522i)11-s + (−0.554 − 0.960i)13-s + 0.485·17-s + 0.917·19-s + (0.834 + 1.44i)23-s + (−0.0999 + 0.173i)25-s + (−0.928 + 1.60i)29-s + (−0.359 − 0.622i)31-s + 0.338·35-s + (0.609 − 1.05i)43-s + (0.583 − 1.01i)47-s + (0.214 + 0.371i)49-s − 0.824·53-s + ⋯

Functional equation

Λ(s)=(3240s/2 ΓC(s) L(s)=((0.766+0.642i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3240s/2 ΓC(s+1/2) L(s)=((0.766+0.642i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32403240    =    23⋅34⋅52^{3} \cdot 3^{4} \cdot 5
Sign: 0.766+0.642i0.766 + 0.642i
Analytic conductor: 25.871525.8715
Root analytic conductor: 5.086405.08640
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3240(2161,⋅)\chi_{3240} (2161, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3240, ( :1/2), 0.766+0.642i)(2,\ 3240,\ (\ :1/2),\ 0.766 + 0.642i)

Particular Values

L(1)L(1) ≈\approx 1.5689797601.568979760
L(12)L(\frac12) ≈\approx 1.5689797601.568979760
L(32)L(\frac{3}{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 1
5 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
good7 1+(1−1.73i)T+(−3.5−6.06i)T2 1 + (1 - 1.73i)T + (-3.5 - 6.06i)T^{2}
11 1+(−1+1.73i)T+(−5.5−9.52i)T2 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2}
13 1+(2+3.46i)T+(−6.5+11.2i)T2 1 + (2 + 3.46i)T + (-6.5 + 11.2i)T^{2}
17 1−2T+17T2 1 - 2T + 17T^{2}
19 1−4T+19T2 1 - 4T + 19T^{2}
23 1+(−4−6.92i)T+(−11.5+19.9i)T2 1 + (-4 - 6.92i)T + (-11.5 + 19.9i)T^{2}
29 1+(5−8.66i)T+(−14.5−25.1i)T2 1 + (5 - 8.66i)T + (-14.5 - 25.1i)T^{2}
31 1+(2+3.46i)T+(−15.5+26.8i)T2 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2}
37 1+37T2 1 + 37T^{2}
41 1+(−20.5+35.5i)T2 1 + (-20.5 + 35.5i)T^{2}
43 1+(−4+6.92i)T+(−21.5−37.2i)T2 1 + (-4 + 6.92i)T + (-21.5 - 37.2i)T^{2}
47 1+(−4+6.92i)T+(−23.5−40.7i)T2 1 + (-4 + 6.92i)T + (-23.5 - 40.7i)T^{2}
53 1+6T+53T2 1 + 6T + 53T^{2}
59 1+(7+12.1i)T+(−29.5+51.0i)T2 1 + (7 + 12.1i)T + (-29.5 + 51.0i)T^{2}
61 1+(−7+12.1i)T+(−30.5−52.8i)T2 1 + (-7 + 12.1i)T + (-30.5 - 52.8i)T^{2}
67 1+(−2−3.46i)T+(−33.5+58.0i)T2 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2}
71 1+12T+71T2 1 + 12T + 71T^{2}
73 1−6T+73T2 1 - 6T + 73T^{2}
79 1+(−6+10.3i)T+(−39.5−68.4i)T2 1 + (-6 + 10.3i)T + (-39.5 - 68.4i)T^{2}
83 1+(−2+3.46i)T+(−41.5−71.8i)T2 1 + (-2 + 3.46i)T + (-41.5 - 71.8i)T^{2}
89 1−12T+89T2 1 - 12T + 89T^{2}
97 1+(−7+12.1i)T+(−48.5−84.0i)T2 1 + (-7 + 12.1i)T + (-48.5 - 84.0i)T^{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

−8.742890953810567990407208661612, −7.63004399668099205975332266263, −7.37451708309472897532855515521, −6.15890544092952890365779185306, −5.39802955626714196967458616118, −5.04366737560992184796880257472, −3.48974079891533671451568922716, −3.26061608063526723867505453473, −1.87727151863891096223848118861, −0.62823639313851453459713161561, 0.915317226796472180235243374418, 2.24450231914837688340994498212, 3.16561483306824179826255778109, 4.15047594691703027703477778111, 4.66963632843920062930073822790, 5.83200602809844329235036109013, 6.62723176863843212535958603291, 7.29406430226838872347904809550, 7.71447420287431932324308901774, 8.869462372282295202957172107483

Graph of the ZZ-function along the critical line