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Properties

Label 2-702-117.103-c1-0-3
Degree 22
Conductor 702702
Sign −0.641−0.767i-0.641 - 0.767i
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 + (0.899 + 3.49i)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 + (0.249 + 0.968i)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)=((−0.641−0.767i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.641 - 0.767i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(702s/2 ΓC(s+1/2) L(s)=((−0.641−0.767i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.641 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 702702    =    2⋅33⋅132 \cdot 3^{3} \cdot 13
Sign: −0.641−0.767i-0.641 - 0.767i
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), −0.641−0.767i)(2,\ 702,\ (\ :1/2),\ -0.641 - 0.767i)

Particular Values

L(1)L(1) ≈\approx 0.407112+0.871286i0.407112 + 0.871286i
L(12)L(\frac12) ≈\approx 0.407112+0.871286i0.407112 + 0.871286i
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+(−0.899−3.49i)T 1 + (-0.899 - 3.49i)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

−10.52794915969822138417053616229, −9.832409976800655215704673972207, −9.175635848411388464977396838884, −8.289681127929790323054570037373, −7.00772390694493551994985162799, −6.49895219756482589786015067519, −5.70601431810311604343934040099, −4.49386642889560195143422629747, −2.70489697350972574092884683555, −1.94451026270754116115691841470, 0.57594786541199468360555219392, 2.13320593943312644301218589724, 3.14796133736459611025735572452, 4.71892050722427052240396350425, 5.68686553089576393525578201113, 6.57045572346122123847081573600, 7.69450803199012722426607267413, 8.736027005064463933233138385309, 9.214326476850437101505597483271, 10.16900123357602207237185726630

Graph of the ZZ-function along the critical line