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Properties

Label 2-2e4-16.13-c3-0-3
Degree 22
Conductor 1616
Sign 0.690+0.723i0.690 + 0.723i
Analytic cond. 0.9440300.944030
Root an. cond. 0.9716120.971612
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  + (0.836 − 2.70i)2-s + (1.98 + 1.98i)3-s + (−6.59 − 4.52i)4-s + (−0.596 + 0.596i)5-s + (7.01 − 3.69i)6-s + 29.0i·7-s + (−17.7 + 14.0i)8-s − 19.1i·9-s + (1.11 + 2.11i)10-s + (12.1 − 12.1i)11-s + (−4.11 − 22.0i)12-s + (−48.5 − 48.5i)13-s + (78.5 + 24.3i)14-s − 2.36·15-s + (23.0 + 59.6i)16-s + 86.7·17-s + ⋯
L(s)  = 1  + (0.295 − 0.955i)2-s + (0.381 + 0.381i)3-s + (−0.824 − 0.565i)4-s + (−0.0533 + 0.0533i)5-s + (0.477 − 0.251i)6-s + 1.57i·7-s + (−0.784 + 0.620i)8-s − 0.708i·9-s + (0.0351 + 0.0667i)10-s + (0.332 − 0.332i)11-s + (−0.0991 − 0.530i)12-s + (−1.03 − 1.03i)13-s + (1.50 + 0.464i)14-s − 0.0407·15-s + (0.360 + 0.932i)16-s + 1.23·17-s + ⋯

Functional equation

Λ(s)=(16s/2 ΓC(s) L(s)=((0.690+0.723i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.690 + 0.723i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(16s/2 ΓC(s+3/2) L(s)=((0.690+0.723i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.690 + 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1616    =    242^{4}
Sign: 0.690+0.723i0.690 + 0.723i
Analytic conductor: 0.9440300.944030
Root analytic conductor: 0.9716120.971612
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ16(13,⋅)\chi_{16} (13, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 16, ( :3/2), 0.690+0.723i)(2,\ 16,\ (\ :3/2),\ 0.690 + 0.723i)

Particular Values

L(2)L(2) ≈\approx 1.05836−0.452995i1.05836 - 0.452995i
L(12)L(\frac12) ≈\approx 1.05836−0.452995i1.05836 - 0.452995i
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+(−0.836+2.70i)T 1 + (-0.836 + 2.70i)T
good3 1+(−1.98−1.98i)T+27iT2 1 + (-1.98 - 1.98i)T + 27iT^{2}
5 1+(0.596−0.596i)T−125iT2 1 + (0.596 - 0.596i)T - 125iT^{2}
7 1−29.0iT−343T2 1 - 29.0iT - 343T^{2}
11 1+(−12.1+12.1i)T−1.33e3iT2 1 + (-12.1 + 12.1i)T - 1.33e3iT^{2}
13 1+(48.5+48.5i)T+2.19e3iT2 1 + (48.5 + 48.5i)T + 2.19e3iT^{2}
17 1−86.7T+4.91e3T2 1 - 86.7T + 4.91e3T^{2}
19 1+(54.8+54.8i)T+6.85e3iT2 1 + (54.8 + 54.8i)T + 6.85e3iT^{2}
23 1−70.2iT−1.21e4T2 1 - 70.2iT - 1.21e4T^{2}
29 1+(−63.4−63.4i)T+2.43e4iT2 1 + (-63.4 - 63.4i)T + 2.43e4iT^{2}
31 1+8.86T+2.97e4T2 1 + 8.86T + 2.97e4T^{2}
37 1+(21.7−21.7i)T−5.06e4iT2 1 + (21.7 - 21.7i)T - 5.06e4iT^{2}
41 1+153.iT−6.89e4T2 1 + 153. iT - 6.89e4T^{2}
43 1+(120.−120.i)T−7.95e4iT2 1 + (120. - 120. i)T - 7.95e4iT^{2}
47 1+99.9T+1.03e5T2 1 + 99.9T + 1.03e5T^{2}
53 1+(−389.+389.i)T−1.48e5iT2 1 + (-389. + 389. i)T - 1.48e5iT^{2}
59 1+(324.−324.i)T−2.05e5iT2 1 + (324. - 324. i)T - 2.05e5iT^{2}
61 1+(0.339+0.339i)T+2.26e5iT2 1 + (0.339 + 0.339i)T + 2.26e5iT^{2}
67 1+(−565.−565.i)T+3.00e5iT2 1 + (-565. - 565. i)T + 3.00e5iT^{2}
71 1+419.iT−3.57e5T2 1 + 419. iT - 3.57e5T^{2}
73 1−374.iT−3.89e5T2 1 - 374. iT - 3.89e5T^{2}
79 1+705.T+4.93e5T2 1 + 705.T + 4.93e5T^{2}
83 1+(947.+947.i)T+5.71e5iT2 1 + (947. + 947. i)T + 5.71e5iT^{2}
89 1+4.72iT−7.04e5T2 1 + 4.72iT - 7.04e5T^{2}
97 1−379.T+9.12e5T2 1 - 379.T + 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

−18.81499708562010550193179178208, −17.58272052045953503232993107530, −15.30440907487681536633485048095, −14.62756423135920295629486760238, −12.70323518409211395850602867151, −11.73893113873026473213007782786, −9.874240689495280735030158463244, −8.742826986623313927493260323542, −5.47840845349865906440710770840, −3.06015101561518229796816170911, 4.40303099004854676081851310537, 6.92156721859609382299166795991, 8.013605696082205649768765020321, 10.05390139051174890333001209254, 12.41873289532976287133991246237, 13.88395681024504428049634726154, 14.47141120147056739317482069840, 16.53554221390283811333753072371, 17.02127008544090049789418520416, 18.78089501296863540170105465116

Graph of the ZZ-function along the critical line