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Properties

Label 2-702-117.103-c1-0-6
Degree 22
Conductor 702702
Sign 0.970+0.241i0.970 + 0.241i
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 + (0.419 + 0.242i)5-s + (4.37 − 2.52i)7-s + 0.999i·8-s − 0.484·10-s + (2.78 − 1.60i)11-s + (−0.722 + 3.53i)13-s + (−2.52 + 4.37i)14-s + (−0.5 − 0.866i)16-s − 4.20·17-s − 3.21i·19-s + (0.419 − 0.242i)20-s + (−1.60 + 2.78i)22-s + (3.13 − 5.43i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (0.187 + 0.108i)5-s + (1.65 − 0.955i)7-s + 0.353i·8-s − 0.153·10-s + (0.838 − 0.484i)11-s + (−0.200 + 0.979i)13-s + (−0.675 + 1.16i)14-s + (−0.125 − 0.216i)16-s − 1.01·17-s − 0.736i·19-s + (0.0937 − 0.0541i)20-s + (−0.342 + 0.592i)22-s + (0.653 − 1.13i)23-s + ⋯

Functional equation

Λ(s)=(702s/2 ΓC(s) L(s)=((0.970+0.241i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(702s/2 ΓC(s+1/2) L(s)=((0.970+0.241i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 702702    =    2⋅33⋅132 \cdot 3^{3} \cdot 13
Sign: 0.970+0.241i0.970 + 0.241i
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), 0.970+0.241i)(2,\ 702,\ (\ :1/2),\ 0.970 + 0.241i)

Particular Values

L(1)L(1) ≈\approx 1.39268−0.170709i1.39268 - 0.170709i
L(12)L(\frac12) ≈\approx 1.39268−0.170709i1.39268 - 0.170709i
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+(0.722−3.53i)T 1 + (0.722 - 3.53i)T
good5 1+(−0.419−0.242i)T+(2.5+4.33i)T2 1 + (-0.419 - 0.242i)T + (2.5 + 4.33i)T^{2}
7 1+(−4.37+2.52i)T+(3.5−6.06i)T2 1 + (-4.37 + 2.52i)T + (3.5 - 6.06i)T^{2}
11 1+(−2.78+1.60i)T+(5.5−9.52i)T2 1 + (-2.78 + 1.60i)T + (5.5 - 9.52i)T^{2}
17 1+4.20T+17T2 1 + 4.20T + 17T^{2}
19 1+3.21iT−19T2 1 + 3.21iT - 19T^{2}
23 1+(−3.13+5.43i)T+(−11.5−19.9i)T2 1 + (-3.13 + 5.43i)T + (-11.5 - 19.9i)T^{2}
29 1+(−2.29−3.97i)T+(−14.5+25.1i)T2 1 + (-2.29 - 3.97i)T + (-14.5 + 25.1i)T^{2}
31 1+(5.61+3.24i)T+(15.5+26.8i)T2 1 + (5.61 + 3.24i)T + (15.5 + 26.8i)T^{2}
37 1−2.08iT−37T2 1 - 2.08iT - 37T^{2}
41 1+(−9.57−5.52i)T+(20.5+35.5i)T2 1 + (-9.57 - 5.52i)T + (20.5 + 35.5i)T^{2}
43 1+(−4.73−8.19i)T+(−21.5+37.2i)T2 1 + (-4.73 - 8.19i)T + (-21.5 + 37.2i)T^{2}
47 1+(−4.57+2.64i)T+(23.5−40.7i)T2 1 + (-4.57 + 2.64i)T + (23.5 - 40.7i)T^{2}
53 1+6.41T+53T2 1 + 6.41T + 53T^{2}
59 1+(3.13+1.81i)T+(29.5+51.0i)T2 1 + (3.13 + 1.81i)T + (29.5 + 51.0i)T^{2}
61 1+(−0.500−0.867i)T+(−30.5+52.8i)T2 1 + (-0.500 - 0.867i)T + (-30.5 + 52.8i)T^{2}
67 1+(−0.936−0.540i)T+(33.5+58.0i)T2 1 + (-0.936 - 0.540i)T + (33.5 + 58.0i)T^{2}
71 1−4.63iT−71T2 1 - 4.63iT - 71T^{2}
73 1−0.325iT−73T2 1 - 0.325iT - 73T^{2}
79 1+(−3.91−6.78i)T+(−39.5+68.4i)T2 1 + (-3.91 - 6.78i)T + (-39.5 + 68.4i)T^{2}
83 1+(−5.08+2.93i)T+(41.5−71.8i)T2 1 + (-5.08 + 2.93i)T + (41.5 - 71.8i)T^{2}
89 1+8.42iT−89T2 1 + 8.42iT - 89T^{2}
97 1+(−11.3+6.52i)T+(48.5−84.0i)T2 1 + (-11.3 + 6.52i)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

−10.55115825042285957301219495649, −9.317669205473576912685329994291, −8.723665841064165268165160527011, −7.83113947902049636025934728816, −6.97490644492772586255266064848, −6.26055491307034449638073056835, −4.78215110319945908896926000317, −4.23900068862688850853186852712, −2.27243999679749392414225407402, −1.06049422694143909365559047265, 1.47047293553300652548019968994, 2.34704882453362132474654478047, 3.90782589901344218958458761530, 5.09583940665502503909231224207, 5.88972106993174284694283563866, 7.35609883921442050883427732217, 7.931378990562882695114888169486, 8.999952480688752053747129538529, 9.312170843855763427243031679413, 10.67004961405840192959325130697

Graph of the ZZ-function along the critical line