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Properties

Label 2-960-240.29-c0-0-1
Degree 22
Conductor 960960
Sign 0.923−0.382i0.923 - 0.382i
Analytic cond. 0.4791020.479102
Root an. cond. 0.6921720.692172
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 0.707i)3-s + (0.707 − 0.707i)5-s + 1.00i·9-s + 1.00·15-s − 1.41·17-s + (1 + i)19-s − 1.41i·23-s − 1.00i·25-s + (−0.707 + 0.707i)27-s + (0.707 + 0.707i)45-s − 1.41·47-s − 49-s + (−1.00 − 1.00i)51-s + 1.41i·57-s + (1 + i)61-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)3-s + (0.707 − 0.707i)5-s + 1.00i·9-s + 1.00·15-s − 1.41·17-s + (1 + i)19-s − 1.41i·23-s − 1.00i·25-s + (−0.707 + 0.707i)27-s + (0.707 + 0.707i)45-s − 1.41·47-s − 49-s + (−1.00 − 1.00i)51-s + 1.41i·57-s + (1 + i)61-s + ⋯

Functional equation

Λ(s)=(960s/2 ΓC(s) L(s)=((0.923−0.382i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(960s/2 ΓC(s) L(s)=((0.923−0.382i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 960960    =    26⋅3⋅52^{6} \cdot 3 \cdot 5
Sign: 0.923−0.382i0.923 - 0.382i
Analytic conductor: 0.4791020.479102
Root analytic conductor: 0.6921720.692172
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ960(209,⋅)\chi_{960} (209, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 960, ( :0), 0.923−0.382i)(2,\ 960,\ (\ :0),\ 0.923 - 0.382i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.3750479861.375047986
L(12)L(\frac12) ≈\approx 1.3750479861.375047986
L(1)L(1) 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+(−0.707−0.707i)T 1 + (-0.707 - 0.707i)T
5 1+(−0.707+0.707i)T 1 + (-0.707 + 0.707i)T
good7 1+T2 1 + T^{2}
11 1+iT2 1 + iT^{2}
13 1−iT2 1 - iT^{2}
17 1+1.41T+T2 1 + 1.41T + T^{2}
19 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{2}
23 1+1.41iT−T2 1 + 1.41iT - T^{2}
29 1−iT2 1 - iT^{2}
31 1+T2 1 + T^{2}
37 1+iT2 1 + iT^{2}
41 1+T2 1 + T^{2}
43 1+iT2 1 + iT^{2}
47 1+1.41T+T2 1 + 1.41T + T^{2}
53 1−iT2 1 - iT^{2}
59 1+iT2 1 + iT^{2}
61 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{2}
67 1−iT2 1 - iT^{2}
71 1+T2 1 + T^{2}
73 1+T2 1 + T^{2}
79 1+2T+T2 1 + 2T + T^{2}
83 1+iT2 1 + iT^{2}
89 1+T2 1 + T^{2}
97 1−T2 1 - 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

−10.05138144830170383170687185455, −9.496499585742355701194734615427, −8.635544803135460749451788695433, −8.160980986375860629738448735087, −6.89582988152375854335416825106, −5.82863166938249298973048772723, −4.87735690847406969783657677496, −4.16669900258668777363601989195, −2.88035360203597166748712738625, −1.79310339574997505454639337843, 1.65781942631712337322272859052, 2.67837960823390658538611852837, 3.54961597074933272583482833432, 5.00721713175100405242841594570, 6.14003037804188266670919153498, 6.88848049230921251143996625197, 7.49156758015824825761964801315, 8.562263352978267541676388236071, 9.386362882517186432702955875513, 9.894572229778655794612678331111

Graph of the ZZ-function along the critical line