↑

Properties

Label 2-624-52.31-c3-0-34
Degree 22
Conductor 624624
Sign −0.0655+0.997i-0.0655 + 0.997i
Analytic cond. 36.817136.8171
Root an. cond. 6.067716.06771
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3i·3-s + (13.7 − 13.7i)5-s + (−9.00 + 9.00i)7-s − 9·9-s + (5.13 − 5.13i)11-s + (−29.7 − 36.2i)13-s + (41.1 + 41.1i)15-s − 41.0i·17-s + (28.6 + 28.6i)19-s + (−27.0 − 27.0i)21-s + 58.8·23-s − 251. i·25-s − 27i·27-s − 13.6·29-s + (−163. − 163. i)31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (1.22 − 1.22i)5-s + (−0.486 + 0.486i)7-s − 0.333·9-s + (0.140 − 0.140i)11-s + (−0.634 − 0.772i)13-s + (0.708 + 0.708i)15-s − 0.586i·17-s + (0.345 + 0.345i)19-s + (−0.280 − 0.280i)21-s + 0.533·23-s − 2.01i·25-s − 0.192i·27-s − 0.0871·29-s + (−0.949 − 0.949i)31-s + ⋯

Functional equation

Λ(s)=(624s/2 ΓC(s) L(s)=((−0.0655+0.997i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0655 + 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(624s/2 ΓC(s+3/2) L(s)=((−0.0655+0.997i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0655 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 624624    =    24⋅3⋅132^{4} \cdot 3 \cdot 13
Sign: −0.0655+0.997i-0.0655 + 0.997i
Analytic conductor: 36.817136.8171
Root analytic conductor: 6.067716.06771
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ624(31,⋅)\chi_{624} (31, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 624, ( :3/2), −0.0655+0.997i)(2,\ 624,\ (\ :3/2),\ -0.0655 + 0.997i)

Particular Values

L(2)L(2) ≈\approx 1.6910373341.691037334
L(12)L(\frac12) ≈\approx 1.6910373341.691037334
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
13 1+(29.7+36.2i)T 1 + (29.7 + 36.2i)T
good5 1+(−13.7+13.7i)T−125iT2 1 + (-13.7 + 13.7i)T - 125iT^{2}
7 1+(9.00−9.00i)T−343iT2 1 + (9.00 - 9.00i)T - 343iT^{2}
11 1+(−5.13+5.13i)T−1.33e3iT2 1 + (-5.13 + 5.13i)T - 1.33e3iT^{2}
17 1+41.0iT−4.91e3T2 1 + 41.0iT - 4.91e3T^{2}
19 1+(−28.6−28.6i)T+6.85e3iT2 1 + (-28.6 - 28.6i)T + 6.85e3iT^{2}
23 1−58.8T+1.21e4T2 1 - 58.8T + 1.21e4T^{2}
29 1+13.6T+2.43e4T2 1 + 13.6T + 2.43e4T^{2}
31 1+(163.+163.i)T+2.97e4iT2 1 + (163. + 163. i)T + 2.97e4iT^{2}
37 1+(129.+129.i)T+5.06e4iT2 1 + (129. + 129. i)T + 5.06e4iT^{2}
41 1+(−30.4+30.4i)T−6.89e4iT2 1 + (-30.4 + 30.4i)T - 6.89e4iT^{2}
43 1+342.T+7.95e4T2 1 + 342.T + 7.95e4T^{2}
47 1+(−328.+328.i)T−1.03e5iT2 1 + (-328. + 328. i)T - 1.03e5iT^{2}
53 1−109.T+1.48e5T2 1 - 109.T + 1.48e5T^{2}
59 1+(262.−262.i)T−2.05e5iT2 1 + (262. - 262. i)T - 2.05e5iT^{2}
61 1−69.0T+2.26e5T2 1 - 69.0T + 2.26e5T^{2}
67 1+(138.+138.i)T+3.00e5iT2 1 + (138. + 138. i)T + 3.00e5iT^{2}
71 1+(185.+185.i)T+3.57e5iT2 1 + (185. + 185. i)T + 3.57e5iT^{2}
73 1+(597.+597.i)T+3.89e5iT2 1 + (597. + 597. i)T + 3.89e5iT^{2}
79 1+627.iT−4.93e5T2 1 + 627. iT - 4.93e5T^{2}
83 1+(−542.−542.i)T+5.71e5iT2 1 + (-542. - 542. i)T + 5.71e5iT^{2}
89 1+(306.+306.i)T+7.04e5iT2 1 + (306. + 306. i)T + 7.04e5iT^{2}
97 1+(−1.22e3+1.22e3i)T−9.12e5iT2 1 + (-1.22e3 + 1.22e3i)T - 9.12e5iT^{2}
show more
show less
   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

−9.792640400056007369723579143549, −9.223086490333367589612988149037, −8.616461600497029111811813273266, −7.35519398465778891437753888925, −5.92096944728621968564785364345, −5.46906053493843936499865424707, −4.60657004335798388775960398231, −3.16060199660410751102279785536, −1.96147963979710766711543701854, −0.46223584914440797692589799938, 1.50378968500484842761152958879, 2.51898618449088654000371279958, 3.52914279873816417128053973270, 5.11647426442156891306982492757, 6.21858057099884688307364656625, 6.85227851393895268209196009888, 7.37941451074813998292974319321, 8.858569986799420875549055404987, 9.691786072537563676170887815592, 10.38606883862940723633403030941

Graph of the ZZ-function along the critical line