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Properties

Label 2-960-80.27-c1-0-3
Degree 22
Conductor 960960
Sign 0.0290−0.999i0.0290 - 0.999i
Analytic cond. 7.665637.66563
Root an. cond. 2.768682.76868
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  − 3-s + (−2.13 − 0.658i)5-s + (3.54 + 3.54i)7-s + 9-s + (0.707 − 0.707i)11-s − 1.18i·13-s + (2.13 + 0.658i)15-s + (−2.63 − 2.63i)17-s + (−5.21 + 5.21i)19-s + (−3.54 − 3.54i)21-s + (1.86 − 1.86i)23-s + (4.13 + 2.81i)25-s − 27-s + (2.17 + 2.17i)29-s + 2.39i·31-s + ⋯
L(s)  = 1  − 0.577·3-s + (−0.955 − 0.294i)5-s + (1.34 + 1.34i)7-s + 0.333·9-s + (0.213 − 0.213i)11-s − 0.329i·13-s + (0.551 + 0.170i)15-s + (−0.639 − 0.639i)17-s + (−1.19 + 1.19i)19-s + (−0.774 − 0.774i)21-s + (0.388 − 0.388i)23-s + (0.826 + 0.562i)25-s − 0.192·27-s + (0.403 + 0.403i)29-s + 0.430i·31-s + ⋯

Functional equation

Λ(s)=(960s/2 ΓC(s) L(s)=((0.0290−0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0290 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(960s/2 ΓC(s+1/2) L(s)=((0.0290−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0290 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 960960    =    26⋅3⋅52^{6} \cdot 3 \cdot 5
Sign: 0.0290−0.999i0.0290 - 0.999i
Analytic conductor: 7.665637.66563
Root analytic conductor: 2.768682.76868
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ960(847,⋅)\chi_{960} (847, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 960, ( :1/2), 0.0290−0.999i)(2,\ 960,\ (\ :1/2),\ 0.0290 - 0.999i)

Particular Values

L(1)L(1) ≈\approx 0.702965+0.682808i0.702965 + 0.682808i
L(12)L(\frac12) ≈\approx 0.702965+0.682808i0.702965 + 0.682808i
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
3 1+T 1 + T
5 1+(2.13+0.658i)T 1 + (2.13 + 0.658i)T
good7 1+(−3.54−3.54i)T+7iT2 1 + (-3.54 - 3.54i)T + 7iT^{2}
11 1+(−0.707+0.707i)T−11iT2 1 + (-0.707 + 0.707i)T - 11iT^{2}
13 1+1.18iT−13T2 1 + 1.18iT - 13T^{2}
17 1+(2.63+2.63i)T+17iT2 1 + (2.63 + 2.63i)T + 17iT^{2}
19 1+(5.21−5.21i)T−19iT2 1 + (5.21 - 5.21i)T - 19iT^{2}
23 1+(−1.86+1.86i)T−23iT2 1 + (-1.86 + 1.86i)T - 23iT^{2}
29 1+(−2.17−2.17i)T+29iT2 1 + (-2.17 - 2.17i)T + 29iT^{2}
31 1−2.39iT−31T2 1 - 2.39iT - 31T^{2}
37 1−0.910iT−37T2 1 - 0.910iT - 37T^{2}
41 1−8.26iT−41T2 1 - 8.26iT - 41T^{2}
43 1−10.6iT−43T2 1 - 10.6iT - 43T^{2}
47 1+(5.06−5.06i)T−47iT2 1 + (5.06 - 5.06i)T - 47iT^{2}
53 1−3.52T+53T2 1 - 3.52T + 53T^{2}
59 1+(−10.2−10.2i)T+59iT2 1 + (-10.2 - 10.2i)T + 59iT^{2}
61 1+(−4.49+4.49i)T−61iT2 1 + (-4.49 + 4.49i)T - 61iT^{2}
67 1+1.27iT−67T2 1 + 1.27iT - 67T^{2}
71 1−3.56T+71T2 1 - 3.56T + 71T^{2}
73 1+(2.47+2.47i)T+73iT2 1 + (2.47 + 2.47i)T + 73iT^{2}
79 1+3.89T+79T2 1 + 3.89T + 79T^{2}
83 1+9.99T+83T2 1 + 9.99T + 83T^{2}
89 1+5.16T+89T2 1 + 5.16T + 89T^{2}
97 1+(6.87+6.87i)T+97iT2 1 + (6.87 + 6.87i)T + 97iT^{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

−10.46022950821491630864294424224, −9.201018751058938840151545096614, −8.361733220049740518000602141114, −8.036219004958736205415490066231, −6.78338416934721774952343017891, −5.81028351176249481920489242236, −4.90884346647219467349208546789, −4.31229645912697937378101752773, −2.79321338477246862840702057915, −1.39933812597711903030519505507, 0.53321099982145651262334543614, 2.03793537146418337708930142386, 3.95559240918277387577727644100, 4.29585340501749025209866462823, 5.25569458195805703202851457924, 6.91623294398779808820450188192, 6.96847897032918366178493866647, 8.127251241270472680923325684402, 8.745667376144678671621416218053, 10.18332605387011223852595550319

Graph of the ZZ-function along the critical line