Properties

Label 2-42-21.17-c5-0-6
Degree 22
Conductor 4242
Sign 0.7630.645i0.763 - 0.645i
Analytic cond. 6.736126.73612
Root an. cond. 2.595402.59540
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.46 − 2i)2-s + (9.95 + 11.9i)3-s + (7.99 − 13.8i)4-s + (41.5 + 72.0i)5-s + (58.4 + 21.6i)6-s + (−128. − 15.2i)7-s − 63.9i·8-s + (−44.9 + 238. i)9-s + (288. + 166. i)10-s + (455. + 263. i)11-s + (245. − 41.9i)12-s − 776. i·13-s + (−476. + 204. i)14-s + (−450. + 1.21e3i)15-s + (−128 − 221. i)16-s + (236. − 409. i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.638 + 0.769i)3-s + (0.249 − 0.433i)4-s + (0.744 + 1.28i)5-s + (0.663 + 0.245i)6-s + (−0.993 − 0.117i)7-s − 0.353i·8-s + (−0.184 + 0.982i)9-s + (0.911 + 0.526i)10-s + (1.13 + 0.655i)11-s + (0.492 − 0.0839i)12-s − 1.27i·13-s + (−0.649 + 0.279i)14-s + (−0.516 + 1.39i)15-s + (−0.125 − 0.216i)16-s + (0.198 − 0.343i)17-s + ⋯

Functional equation

Λ(s)=(42s/2ΓC(s)L(s)=((0.7630.645i)Λ(6s)\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.645i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(42s/2ΓC(s+5/2)L(s)=((0.7630.645i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 4242    =    2372 \cdot 3 \cdot 7
Sign: 0.7630.645i0.763 - 0.645i
Analytic conductor: 6.736126.73612
Root analytic conductor: 2.595402.59540
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ42(17,)\chi_{42} (17, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 42, ( :5/2), 0.7630.645i)(2,\ 42,\ (\ :5/2),\ 0.763 - 0.645i)

Particular Values

L(3)L(3) \approx 2.63582+0.964384i2.63582 + 0.964384i
L(12)L(\frac12) \approx 2.63582+0.964384i2.63582 + 0.964384i
L(72)L(\frac{7}{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+(3.46+2i)T 1 + (-3.46 + 2i)T
3 1+(9.9511.9i)T 1 + (-9.95 - 11.9i)T
7 1+(128.+15.2i)T 1 + (128. + 15.2i)T
good5 1+(41.572.0i)T+(1.56e3+2.70e3i)T2 1 + (-41.5 - 72.0i)T + (-1.56e3 + 2.70e3i)T^{2}
11 1+(455.263.i)T+(8.05e4+1.39e5i)T2 1 + (-455. - 263. i)T + (8.05e4 + 1.39e5i)T^{2}
13 1+776.iT3.71e5T2 1 + 776. iT - 3.71e5T^{2}
17 1+(236.+409.i)T+(7.09e51.22e6i)T2 1 + (-236. + 409. i)T + (-7.09e5 - 1.22e6i)T^{2}
19 1+(1.71e3+990.i)T+(1.23e62.14e6i)T2 1 + (-1.71e3 + 990. i)T + (1.23e6 - 2.14e6i)T^{2}
23 1+(1.49e3863.i)T+(3.21e65.57e6i)T2 1 + (1.49e3 - 863. i)T + (3.21e6 - 5.57e6i)T^{2}
29 1+6.04e3iT2.05e7T2 1 + 6.04e3iT - 2.05e7T^{2}
31 1+(7.50e3+4.33e3i)T+(1.43e7+2.47e7i)T2 1 + (7.50e3 + 4.33e3i)T + (1.43e7 + 2.47e7i)T^{2}
37 1+(1.50e32.61e3i)T+(3.46e7+6.00e7i)T2 1 + (-1.50e3 - 2.61e3i)T + (-3.46e7 + 6.00e7i)T^{2}
41 1+1.63e3T+1.15e8T2 1 + 1.63e3T + 1.15e8T^{2}
43 1+1.19e3T+1.47e8T2 1 + 1.19e3T + 1.47e8T^{2}
47 1+(1.93e33.35e3i)T+(1.14e8+1.98e8i)T2 1 + (-1.93e3 - 3.35e3i)T + (-1.14e8 + 1.98e8i)T^{2}
53 1+(3.24e31.87e3i)T+(2.09e8+3.62e8i)T2 1 + (-3.24e3 - 1.87e3i)T + (2.09e8 + 3.62e8i)T^{2}
59 1+(3.24e3+5.61e3i)T+(3.57e86.19e8i)T2 1 + (-3.24e3 + 5.61e3i)T + (-3.57e8 - 6.19e8i)T^{2}
61 1+(3.85e4+2.22e4i)T+(4.22e87.31e8i)T2 1 + (-3.85e4 + 2.22e4i)T + (4.22e8 - 7.31e8i)T^{2}
67 1+(1.21e42.11e4i)T+(6.75e81.16e9i)T2 1 + (1.21e4 - 2.11e4i)T + (-6.75e8 - 1.16e9i)T^{2}
71 18.35e4iT1.80e9T2 1 - 8.35e4iT - 1.80e9T^{2}
73 1+(1.11e46.41e3i)T+(1.03e9+1.79e9i)T2 1 + (-1.11e4 - 6.41e3i)T + (1.03e9 + 1.79e9i)T^{2}
79 1+(2.01e4+3.49e4i)T+(1.53e9+2.66e9i)T2 1 + (2.01e4 + 3.49e4i)T + (-1.53e9 + 2.66e9i)T^{2}
83 1+7.28e4T+3.93e9T2 1 + 7.28e4T + 3.93e9T^{2}
89 1+(2.92e4+5.07e4i)T+(2.79e9+4.83e9i)T2 1 + (2.92e4 + 5.07e4i)T + (-2.79e9 + 4.83e9i)T^{2}
97 1+1.01e3iT8.58e9T2 1 + 1.01e3iT - 8.58e9T^{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

−14.92222019511902153311385666716, −14.06551645243701500054580938720, −13.12682863665083909708274394662, −11.39397248150130743083170646151, −10.04088135940735687188022390389, −9.616589036066281380033118348442, −7.21530191020795344088149084624, −5.76221557846472752893220133722, −3.69473379189226860246875555721, −2.60152868956206793374342082424, 1.51120121513796403814305100814, 3.67600093243620725543663626927, 5.74951920659998844506178953976, 6.87553710736478587349155230822, 8.717680605751621267961412066836, 9.402779986374174262024610751867, 11.98552130732788742357072364665, 12.70226427291572458849210643155, 13.75239060250356885112777821269, 14.40141018863389372915508890586

Graph of the ZZ-function along the critical line