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Properties

Label 2-702-117.83-c1-0-13
Degree 22
Conductor 702702
Sign 0.317+0.948i0.317 + 0.948i
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  + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (0.926 − 3.45i)5-s + (4.42 − 1.18i)7-s + (0.707 − 0.707i)8-s − 3.57i·10-s + (0.277 + 1.03i)11-s + (−3.47 + 0.956i)13-s + (3.96 − 2.29i)14-s + (0.500 − 0.866i)16-s − 4.13·17-s + (−3.55 + 3.55i)19-s + (−0.926 − 3.45i)20-s + (0.536 + 0.928i)22-s + (1.26 + 2.18i)23-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.433 − 0.249i)4-s + (0.414 − 1.54i)5-s + (1.67 − 0.448i)7-s + (0.249 − 0.249i)8-s − 1.13i·10-s + (0.0836 + 0.312i)11-s + (−0.964 + 0.265i)13-s + (1.06 − 0.612i)14-s + (0.125 − 0.216i)16-s − 1.00·17-s + (−0.814 + 0.814i)19-s + (−0.207 − 0.772i)20-s + (0.114 + 0.197i)22-s + (0.263 + 0.456i)23-s + ⋯

Functional equation

Λ(s)=(702s/2 ΓC(s) L(s)=((0.317+0.948i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.317 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(702s/2 ΓC(s+1/2) L(s)=((0.317+0.948i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.317 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 702702    =    2⋅33⋅132 \cdot 3^{3} \cdot 13
Sign: 0.317+0.948i0.317 + 0.948i
Analytic conductor: 5.605495.60549
Root analytic conductor: 2.367592.36759
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ702(629,⋅)\chi_{702} (629, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 702, ( :1/2), 0.317+0.948i)(2,\ 702,\ (\ :1/2),\ 0.317 + 0.948i)

Particular Values

L(1)L(1) ≈\approx 2.18072−1.57029i2.18072 - 1.57029i
L(12)L(\frac12) ≈\approx 2.18072−1.57029i2.18072 - 1.57029i
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+(−0.965+0.258i)T 1 + (-0.965 + 0.258i)T
3 1 1
13 1+(3.47−0.956i)T 1 + (3.47 - 0.956i)T
good5 1+(−0.926+3.45i)T+(−4.33−2.5i)T2 1 + (-0.926 + 3.45i)T + (-4.33 - 2.5i)T^{2}
7 1+(−4.42+1.18i)T+(6.06−3.5i)T2 1 + (-4.42 + 1.18i)T + (6.06 - 3.5i)T^{2}
11 1+(−0.277−1.03i)T+(−9.52+5.5i)T2 1 + (-0.277 - 1.03i)T + (-9.52 + 5.5i)T^{2}
17 1+4.13T+17T2 1 + 4.13T + 17T^{2}
19 1+(3.55−3.55i)T−19iT2 1 + (3.55 - 3.55i)T - 19iT^{2}
23 1+(−1.26−2.18i)T+(−11.5+19.9i)T2 1 + (-1.26 - 2.18i)T + (-11.5 + 19.9i)T^{2}
29 1+(−0.654−0.377i)T+(14.5+25.1i)T2 1 + (-0.654 - 0.377i)T + (14.5 + 25.1i)T^{2}
31 1+(−8.68−2.32i)T+(26.8+15.5i)T2 1 + (-8.68 - 2.32i)T + (26.8 + 15.5i)T^{2}
37 1+(−1.53−1.53i)T+37iT2 1 + (-1.53 - 1.53i)T + 37iT^{2}
41 1+(0.112−0.419i)T+(−35.5−20.5i)T2 1 + (0.112 - 0.419i)T + (-35.5 - 20.5i)T^{2}
43 1+(2.15+1.24i)T+(21.5+37.2i)T2 1 + (2.15 + 1.24i)T + (21.5 + 37.2i)T^{2}
47 1+(0.366+1.36i)T+(−40.7+23.5i)T2 1 + (0.366 + 1.36i)T + (-40.7 + 23.5i)T^{2}
53 1−11.0iT−53T2 1 - 11.0iT - 53T^{2}
59 1+(9.66+2.58i)T+(51.0+29.5i)T2 1 + (9.66 + 2.58i)T + (51.0 + 29.5i)T^{2}
61 1+(−5.37+9.31i)T+(−30.5−52.8i)T2 1 + (-5.37 + 9.31i)T + (-30.5 - 52.8i)T^{2}
67 1+(−8.82−2.36i)T+(58.0+33.5i)T2 1 + (-8.82 - 2.36i)T + (58.0 + 33.5i)T^{2}
71 1+(2.42+2.42i)T+71iT2 1 + (2.42 + 2.42i)T + 71iT^{2}
73 1+(−3.17−3.17i)T+73iT2 1 + (-3.17 - 3.17i)T + 73iT^{2}
79 1+(0.638−1.10i)T+(−39.5−68.4i)T2 1 + (0.638 - 1.10i)T + (-39.5 - 68.4i)T^{2}
83 1+(−6.13+1.64i)T+(71.8−41.5i)T2 1 + (-6.13 + 1.64i)T + (71.8 - 41.5i)T^{2}
89 1+(6.96−6.96i)T−89iT2 1 + (6.96 - 6.96i)T - 89iT^{2}
97 1+(−0.666−2.48i)T+(−84.0+48.5i)T2 1 + (-0.666 - 2.48i)T + (-84.0 + 48.5i)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

−10.37964040982296871745751673789, −9.420509545420442554783363056251, −8.458756174148883351133833629612, −7.82157950368081445736611836155, −6.60817711403094358440856914073, −5.34299986821979403107957070266, −4.69630230058956844223116907302, −4.25798956292361137540752167057, −2.18150016625220709362957930440, −1.30751505199061061093089931271, 2.18671102448849937066007415918, 2.74561273062324700740286356888, 4.34035501850540542721544379086, 5.11618509637169620768378127093, 6.24613142028856740700986932037, 6.92112680232185620611332161374, 7.85761212791000608754576223034, 8.699575505565173007187302695048, 10.01922137179293002388972421976, 10.93711185976064191666638024355

Graph of the ZZ-function along the critical line