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Properties

Label 2-3584-224.69-c0-0-2
Degree 22
Conductor 35843584
Sign 0.555−0.831i0.555 - 0.831i
Analytic cond. 1.788641.78864
Root an. cond. 1.337401.33740
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)7-s + (−0.707 + 0.707i)9-s + (0.292 − 0.707i)11-s + (−1 + i)23-s + (0.707 + 0.707i)25-s + (0.292 + 0.707i)29-s + (1.70 + 0.707i)37-s + (0.707 − 1.70i)43-s + 1.00i·49-s + (−0.707 + 1.70i)53-s − 1.00·63-s + (−0.707 − 1.70i)67-s + (0.707 − 0.292i)77-s + 1.41i·79-s − 1.00i·81-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)9-s + (0.292 − 0.707i)11-s + (−1 + i)23-s + (0.707 + 0.707i)25-s + (0.292 + 0.707i)29-s + (1.70 + 0.707i)37-s + (0.707 − 1.70i)43-s + 1.00i·49-s + (−0.707 + 1.70i)53-s − 1.00·63-s + (−0.707 − 1.70i)67-s + (0.707 − 0.292i)77-s + 1.41i·79-s − 1.00i·81-s + ⋯

Functional equation

Λ(s)=(3584s/2 ΓC(s) L(s)=((0.555−0.831i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3584s/2 ΓC(s) L(s)=((0.555−0.831i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 35843584    =    29⋅72^{9} \cdot 7
Sign: 0.555−0.831i0.555 - 0.831i
Analytic conductor: 1.788641.78864
Root analytic conductor: 1.337401.33740
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3584(3009,⋅)\chi_{3584} (3009, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3584, ( :0), 0.555−0.831i)(2,\ 3584,\ (\ :0),\ 0.555 - 0.831i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.2752189291.275218929
L(12)L(\frac12) ≈\approx 1.2752189291.275218929
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
7 1+(−0.707−0.707i)T 1 + (-0.707 - 0.707i)T
good3 1+(0.707−0.707i)T2 1 + (0.707 - 0.707i)T^{2}
5 1+(−0.707−0.707i)T2 1 + (-0.707 - 0.707i)T^{2}
11 1+(−0.292+0.707i)T+(−0.707−0.707i)T2 1 + (-0.292 + 0.707i)T + (-0.707 - 0.707i)T^{2}
13 1+(−0.707+0.707i)T2 1 + (-0.707 + 0.707i)T^{2}
17 1+T2 1 + T^{2}
19 1+(−0.707+0.707i)T2 1 + (-0.707 + 0.707i)T^{2}
23 1+(1−i)T−iT2 1 + (1 - i)T - iT^{2}
29 1+(−0.292−0.707i)T+(−0.707+0.707i)T2 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2}
31 1−T2 1 - T^{2}
37 1+(−1.70−0.707i)T+(0.707+0.707i)T2 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2}
41 1+iT2 1 + iT^{2}
43 1+(−0.707+1.70i)T+(−0.707−0.707i)T2 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2}
47 1+T2 1 + T^{2}
53 1+(0.707−1.70i)T+(−0.707−0.707i)T2 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2}
59 1+(−0.707−0.707i)T2 1 + (-0.707 - 0.707i)T^{2}
61 1+(0.707−0.707i)T2 1 + (0.707 - 0.707i)T^{2}
67 1+(0.707+1.70i)T+(−0.707+0.707i)T2 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2}
71 1+iT2 1 + iT^{2}
73 1+iT2 1 + iT^{2}
79 1−1.41iT−T2 1 - 1.41iT - T^{2}
83 1+(−0.707+0.707i)T2 1 + (-0.707 + 0.707i)T^{2}
89 1−iT2 1 - iT^{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

−8.935072212308500282125009767967, −8.003055556160490830667970849485, −7.68912260019325639425251617179, −6.49644311024527566743766557059, −5.71237380976683009218298044170, −5.24749622346762657890946317926, −4.32309214743614810578938141672, −3.24167093855471127612065672866, −2.43467870799578720931788950688, −1.38353307970384567398214102261, 0.800750680784287917208046400652, 2.09369384177325243478475484454, 3.04445845460318611117030602416, 4.31243237005727480693327736918, 4.45194238776571933584332476643, 5.74346815093206601266829786547, 6.39539091510184593201753401676, 7.11936918972780292562604534109, 8.039133087192735715250182288606, 8.415833442581956001093067034724

Graph of the ZZ-function along the critical line