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Properties

Label 2-702-117.103-c1-0-1
Degree 22
Conductor 702702
Sign −0.246−0.969i-0.246 - 0.969i
Analytic cond. 5.605495.60549
Root an. cond. 2.367592.36759
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  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−2.40 − 1.38i)5-s + (0.759 − 0.438i)7-s + 0.999i·8-s + 2.77·10-s + (−1.92 + 1.11i)11-s + (−3.20 + 1.64i)13-s + (−0.438 + 0.759i)14-s + (−0.5 − 0.866i)16-s + 3.80·17-s + 2.22i·19-s + (−2.40 + 1.38i)20-s + (1.11 − 1.92i)22-s + (−0.259 + 0.449i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−1.07 − 0.621i)5-s + (0.286 − 0.165i)7-s + 0.353i·8-s + 0.878·10-s + (−0.581 + 0.335i)11-s + (−0.889 + 0.456i)13-s + (−0.117 + 0.202i)14-s + (−0.125 − 0.216i)16-s + 0.923·17-s + 0.510i·19-s + (−0.537 + 0.310i)20-s + (0.237 − 0.411i)22-s + (−0.0541 + 0.0937i)23-s + ⋯

Functional equation

Λ(s)=(702s/2 ΓC(s) L(s)=((−0.246−0.969i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.246 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(702s/2 ΓC(s+1/2) L(s)=((−0.246−0.969i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.246 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 702702    =    2⋅33⋅132 \cdot 3^{3} \cdot 13
Sign: −0.246−0.969i-0.246 - 0.969i
Analytic conductor: 5.605495.60549
Root analytic conductor: 2.367592.36759
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ702(415,⋅)\chi_{702} (415, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 702, ( :1/2), −0.246−0.969i)(2,\ 702,\ (\ :1/2),\ -0.246 - 0.969i)

Particular Values

L(1)L(1) ≈\approx 0.344506+0.443130i0.344506 + 0.443130i
L(12)L(\frac12) ≈\approx 0.344506+0.443130i0.344506 + 0.443130i
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+(0.866−0.5i)T 1 + (0.866 - 0.5i)T
3 1 1
13 1+(3.20−1.64i)T 1 + (3.20 - 1.64i)T
good5 1+(2.40+1.38i)T+(2.5+4.33i)T2 1 + (2.40 + 1.38i)T + (2.5 + 4.33i)T^{2}
7 1+(−0.759+0.438i)T+(3.5−6.06i)T2 1 + (-0.759 + 0.438i)T + (3.5 - 6.06i)T^{2}
11 1+(1.92−1.11i)T+(5.5−9.52i)T2 1 + (1.92 - 1.11i)T + (5.5 - 9.52i)T^{2}
17 1−3.80T+17T2 1 - 3.80T + 17T^{2}
19 1−2.22iT−19T2 1 - 2.22iT - 19T^{2}
23 1+(0.259−0.449i)T+(−11.5−19.9i)T2 1 + (0.259 - 0.449i)T + (-11.5 - 19.9i)T^{2}
29 1+(−3.81−6.60i)T+(−14.5+25.1i)T2 1 + (-3.81 - 6.60i)T + (-14.5 + 25.1i)T^{2}
31 1+(−4.97−2.87i)T+(15.5+26.8i)T2 1 + (-4.97 - 2.87i)T + (15.5 + 26.8i)T^{2}
37 1−11.3iT−37T2 1 - 11.3iT - 37T^{2}
41 1+(3.52+2.03i)T+(20.5+35.5i)T2 1 + (3.52 + 2.03i)T + (20.5 + 35.5i)T^{2}
43 1+(−2.81−4.87i)T+(−21.5+37.2i)T2 1 + (-2.81 - 4.87i)T + (-21.5 + 37.2i)T^{2}
47 1+(−0.920+0.531i)T+(23.5−40.7i)T2 1 + (-0.920 + 0.531i)T + (23.5 - 40.7i)T^{2}
53 1+7.29T+53T2 1 + 7.29T + 53T^{2}
59 1+(−3.52−2.03i)T+(29.5+51.0i)T2 1 + (-3.52 - 2.03i)T + (29.5 + 51.0i)T^{2}
61 1+(3.94+6.83i)T+(−30.5+52.8i)T2 1 + (3.94 + 6.83i)T + (-30.5 + 52.8i)T^{2}
67 1+(5.95+3.43i)T+(33.5+58.0i)T2 1 + (5.95 + 3.43i)T + (33.5 + 58.0i)T^{2}
71 1−15.9iT−71T2 1 - 15.9iT - 71T^{2}
73 1−5.24iT−73T2 1 - 5.24iT - 73T^{2}
79 1+(7.09+12.2i)T+(−39.5+68.4i)T2 1 + (7.09 + 12.2i)T + (-39.5 + 68.4i)T^{2}
83 1+(0.641−0.370i)T+(41.5−71.8i)T2 1 + (0.641 - 0.370i)T + (41.5 - 71.8i)T^{2}
89 1+9.89iT−89T2 1 + 9.89iT - 89T^{2}
97 1+(−13.6+7.86i)T+(48.5−84.0i)T2 1 + (-13.6 + 7.86i)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

−10.45832519603459508032248340698, −9.865393428380420454701886532822, −8.777118382683837440290595110365, −8.002829608861665590162642504106, −7.52995231997384468801113556585, −6.50281958209861140228486161550, −5.11335073383316975370212114198, −4.51452383219060409025141756142, −3.05364822793952507297415027486, −1.31978288459664776512085422969, 0.38932738225904763676449231011, 2.43852987377106539910626599472, 3.33258360579458167637642866364, 4.50557207163373720736384429252, 5.74321276084682557086515964417, 7.04232692954228395086447599999, 7.78508601341078468584921223307, 8.237868092458195274280847947048, 9.449947435508166561552119077367, 10.30440647707921296409740460982

Graph of the ZZ-function along the critical line