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Properties

Label 2-1400-35.4-c1-0-2
Degree 22
Conductor 14001400
Sign −0.208−0.978i-0.208 - 0.978i
Analytic cond. 11.179011.1790
Root an. cond. 3.343503.34350
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)3-s + (0.866 + 2.5i)7-s + (−1 − 1.73i)9-s + (−1 + 1.73i)11-s + (3.46 + 2i)17-s + (−1 − 1.73i)19-s + (0.500 − 2.59i)21-s + (0.866 − 0.5i)23-s + 5i·27-s − 9·29-s + (−2 + 3.46i)31-s + (1.73 − 0.999i)33-s + (−3.46 + 2i)37-s + 41-s + 9i·43-s + ⋯
L(s)  = 1  + (−0.499 − 0.288i)3-s + (0.327 + 0.944i)7-s + (−0.333 − 0.577i)9-s + (−0.301 + 0.522i)11-s + (0.840 + 0.485i)17-s + (−0.229 − 0.397i)19-s + (0.109 − 0.566i)21-s + (0.180 − 0.104i)23-s + 0.962i·27-s − 1.67·29-s + (−0.359 + 0.622i)31-s + (0.301 − 0.174i)33-s + (−0.569 + 0.328i)37-s + 0.156·41-s + 1.37i·43-s + ⋯

Functional equation

Λ(s)=(1400s/2 ΓC(s) L(s)=((−0.208−0.978i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.208 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1400s/2 ΓC(s+1/2) L(s)=((−0.208−0.978i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.208 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14001400    =    23⋅52⋅72^{3} \cdot 5^{2} \cdot 7
Sign: −0.208−0.978i-0.208 - 0.978i
Analytic conductor: 11.179011.1790
Root analytic conductor: 3.343503.34350
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1400(249,⋅)\chi_{1400} (249, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1400, ( :1/2), −0.208−0.978i)(2,\ 1400,\ (\ :1/2),\ -0.208 - 0.978i)

Particular Values

L(1)L(1) ≈\approx 0.84388363700.8438836370
L(12)L(\frac12) ≈\approx 0.84388363700.8438836370
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 1
5 1 1
7 1+(−0.866−2.5i)T 1 + (-0.866 - 2.5i)T
good3 1+(0.866+0.5i)T+(1.5+2.59i)T2 1 + (0.866 + 0.5i)T + (1.5 + 2.59i)T^{2}
11 1+(1−1.73i)T+(−5.5−9.52i)T2 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2}
13 1−13T2 1 - 13T^{2}
17 1+(−3.46−2i)T+(8.5+14.7i)T2 1 + (-3.46 - 2i)T + (8.5 + 14.7i)T^{2}
19 1+(1+1.73i)T+(−9.5+16.4i)T2 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2}
23 1+(−0.866+0.5i)T+(11.5−19.9i)T2 1 + (-0.866 + 0.5i)T + (11.5 - 19.9i)T^{2}
29 1+9T+29T2 1 + 9T + 29T^{2}
31 1+(2−3.46i)T+(−15.5−26.8i)T2 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2}
37 1+(3.46−2i)T+(18.5−32.0i)T2 1 + (3.46 - 2i)T + (18.5 - 32.0i)T^{2}
41 1−T+41T2 1 - T + 41T^{2}
43 1−9iT−43T2 1 - 9iT - 43T^{2}
47 1+(23.5−40.7i)T2 1 + (23.5 - 40.7i)T^{2}
53 1+(−8.66−5i)T+(26.5+45.8i)T2 1 + (-8.66 - 5i)T + (26.5 + 45.8i)T^{2}
59 1+(5−8.66i)T+(−29.5−51.0i)T2 1 + (5 - 8.66i)T + (-29.5 - 51.0i)T^{2}
61 1+(4.5+7.79i)T+(−30.5+52.8i)T2 1 + (4.5 + 7.79i)T + (-30.5 + 52.8i)T^{2}
67 1+(−4.33−2.5i)T+(33.5+58.0i)T2 1 + (-4.33 - 2.5i)T + (33.5 + 58.0i)T^{2}
71 1−14T+71T2 1 - 14T + 71T^{2}
73 1+(10.3+6i)T+(36.5+63.2i)T2 1 + (10.3 + 6i)T + (36.5 + 63.2i)T^{2}
79 1+(−7−12.1i)T+(−39.5+68.4i)T2 1 + (-7 - 12.1i)T + (-39.5 + 68.4i)T^{2}
83 1−11iT−83T2 1 - 11iT - 83T^{2}
89 1+(7.5+12.9i)T+(−44.5+77.0i)T2 1 + (7.5 + 12.9i)T + (-44.5 + 77.0i)T^{2}
97 1−18iT−97T2 1 - 18iT - 97T^{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

−9.638996396282050916486415693151, −9.048138778877038820856969586805, −8.183557032470932569577040534226, −7.33485504776220586487453957317, −6.41358048075850871332563902085, −5.63949061459481675727639200303, −5.02516477771015952698130739756, −3.73283270797228823878172899156, −2.62835528059228636525604196630, −1.41421674611134668641749669216, 0.37116713737588800641613554818, 1.94066432002736558143614523660, 3.35922544540925577761682698651, 4.22494646622992729183394093757, 5.31430870361603401102157181646, 5.73011305717528525436359989637, 7.02291128186673319704000025023, 7.69486275974375710735575818510, 8.412490122837720619156473862869, 9.483535054426076803971367886924

Graph of the ZZ-function along the critical line