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Properties

Label 2-702-117.103-c1-0-12
Degree 22
Conductor 702702
Sign −0.754+0.656i-0.754 + 0.656i
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 + (−2.73 − 1.57i)5-s + (1.36 − 0.787i)7-s − 0.999i·8-s − 3.15·10-s + (3.26 − 1.88i)11-s + (−3.47 + 0.966i)13-s + (0.787 − 1.36i)14-s + (−0.5 − 0.866i)16-s − 7.06·17-s − 3.76i·19-s + (−2.73 + 1.57i)20-s + (1.88 − 3.26i)22-s + (1.84 − 3.20i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−1.22 − 0.706i)5-s + (0.515 − 0.297i)7-s − 0.353i·8-s − 0.998·10-s + (0.983 − 0.567i)11-s + (−0.963 + 0.268i)13-s + (0.210 − 0.364i)14-s + (−0.125 − 0.216i)16-s − 1.71·17-s − 0.864i·19-s + (−0.611 + 0.353i)20-s + (0.401 − 0.695i)22-s + (0.385 − 0.667i)23-s + ⋯

Functional equation

Λ(s)=(702s/2 ΓC(s) L(s)=((−0.754+0.656i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.754 + 0.656i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(702s/2 ΓC(s+1/2) L(s)=((−0.754+0.656i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.754 + 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 702702    =    2⋅33⋅132 \cdot 3^{3} \cdot 13
Sign: −0.754+0.656i-0.754 + 0.656i
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), −0.754+0.656i)(2,\ 702,\ (\ :1/2),\ -0.754 + 0.656i)

Particular Values

L(1)L(1) ≈\approx 0.490799−1.31202i0.490799 - 1.31202i
L(12)L(\frac12) ≈\approx 0.490799−1.31202i0.490799 - 1.31202i
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+(3.47−0.966i)T 1 + (3.47 - 0.966i)T
good5 1+(2.73+1.57i)T+(2.5+4.33i)T2 1 + (2.73 + 1.57i)T + (2.5 + 4.33i)T^{2}
7 1+(−1.36+0.787i)T+(3.5−6.06i)T2 1 + (-1.36 + 0.787i)T + (3.5 - 6.06i)T^{2}
11 1+(−3.26+1.88i)T+(5.5−9.52i)T2 1 + (-3.26 + 1.88i)T + (5.5 - 9.52i)T^{2}
17 1+7.06T+17T2 1 + 7.06T + 17T^{2}
19 1+3.76iT−19T2 1 + 3.76iT - 19T^{2}
23 1+(−1.84+3.20i)T+(−11.5−19.9i)T2 1 + (-1.84 + 3.20i)T + (-11.5 - 19.9i)T^{2}
29 1+(0.109+0.189i)T+(−14.5+25.1i)T2 1 + (0.109 + 0.189i)T + (-14.5 + 25.1i)T^{2}
31 1+(2.65+1.53i)T+(15.5+26.8i)T2 1 + (2.65 + 1.53i)T + (15.5 + 26.8i)T^{2}
37 1+0.292iT−37T2 1 + 0.292iT - 37T^{2}
41 1+(6.39+3.69i)T+(20.5+35.5i)T2 1 + (6.39 + 3.69i)T + (20.5 + 35.5i)T^{2}
43 1+(3.05+5.29i)T+(−21.5+37.2i)T2 1 + (3.05 + 5.29i)T + (-21.5 + 37.2i)T^{2}
47 1+(−6.17+3.56i)T+(23.5−40.7i)T2 1 + (-6.17 + 3.56i)T + (23.5 - 40.7i)T^{2}
53 1−14.4T+53T2 1 - 14.4T + 53T^{2}
59 1+(−9.04−5.22i)T+(29.5+51.0i)T2 1 + (-9.04 - 5.22i)T + (29.5 + 51.0i)T^{2}
61 1+(−3.00−5.19i)T+(−30.5+52.8i)T2 1 + (-3.00 - 5.19i)T + (-30.5 + 52.8i)T^{2}
67 1+(−6.33−3.65i)T+(33.5+58.0i)T2 1 + (-6.33 - 3.65i)T + (33.5 + 58.0i)T^{2}
71 1−0.772iT−71T2 1 - 0.772iT - 71T^{2}
73 1+13.5iT−73T2 1 + 13.5iT - 73T^{2}
79 1+(−6.34−10.9i)T+(−39.5+68.4i)T2 1 + (-6.34 - 10.9i)T + (-39.5 + 68.4i)T^{2}
83 1+(0.314−0.181i)T+(41.5−71.8i)T2 1 + (0.314 - 0.181i)T + (41.5 - 71.8i)T^{2}
89 1+7.06iT−89T2 1 + 7.06iT - 89T^{2}
97 1+(−0.535+0.309i)T+(48.5−84.0i)T2 1 + (-0.535 + 0.309i)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

−10.37920863571558539722788072839, −8.972814549032754432878702266102, −8.639994698830473077778853955343, −7.31309250906743765592858519812, −6.70524298718805046591536548898, −5.21126618528690121408019827091, −4.40466053580730461748279097882, −3.84385269843302539214485608928, −2.28785130235375012927598741352, −0.59160368427326070866333136448, 2.15308087925642188568903913339, 3.49846457692906457827039923454, 4.30328210730137559322303065252, 5.20895727514860831828902970257, 6.60423965562610756971703944516, 7.13500838047298436365972733380, 7.970416556852161978893408401167, 8.841659882559981657169030489212, 9.991334451053739728917047695514, 11.15922694279929086543358276989

Graph of the ZZ-function along the critical line