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Properties

Label 2-960-80.3-c1-0-14
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(943,⋅)\chi_{960} (943, \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

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

Graph of the ZZ-function along the critical line