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Properties

Label 2-48e2-8.5-c1-0-0
Degree 22
Conductor 23042304
Sign −0.707−0.707i-0.707 - 0.707i
Analytic cond. 18.397518.3975
Root an. cond. 4.289234.28923
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  + 2i·5-s − 4i·11-s + 2i·13-s − 2·17-s + 4i·19-s − 8·23-s + 25-s + 6i·29-s + 8·31-s + 6i·37-s − 6·41-s + 4i·43-s − 7·49-s + 2i·53-s + 8·55-s + ⋯
L(s)  = 1  + 0.894i·5-s − 1.20i·11-s + 0.554i·13-s − 0.485·17-s + 0.917i·19-s − 1.66·23-s + 0.200·25-s + 1.11i·29-s + 1.43·31-s + 0.986i·37-s − 0.937·41-s + 0.609i·43-s − 49-s + 0.274i·53-s + 1.07·55-s + ⋯

Functional equation

Λ(s)=(2304s/2 ΓC(s) L(s)=((−0.707−0.707i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2304s/2 ΓC(s+1/2) L(s)=((−0.707−0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 23042304    =    28⋅322^{8} \cdot 3^{2}
Sign: −0.707−0.707i-0.707 - 0.707i
Analytic conductor: 18.397518.3975
Root analytic conductor: 4.289234.28923
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2304(1153,⋅)\chi_{2304} (1153, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2304, ( :1/2), −0.707−0.707i)(2,\ 2304,\ (\ :1/2),\ -0.707 - 0.707i)

Particular Values

L(1)L(1) ≈\approx 0.97326842110.9732684211
L(12)L(\frac12) ≈\approx 0.97326842110.9732684211
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 1
good5 1−2iT−5T2 1 - 2iT - 5T^{2}
7 1+7T2 1 + 7T^{2}
11 1+4iT−11T2 1 + 4iT - 11T^{2}
13 1−2iT−13T2 1 - 2iT - 13T^{2}
17 1+2T+17T2 1 + 2T + 17T^{2}
19 1−4iT−19T2 1 - 4iT - 19T^{2}
23 1+8T+23T2 1 + 8T + 23T^{2}
29 1−6iT−29T2 1 - 6iT - 29T^{2}
31 1−8T+31T2 1 - 8T + 31T^{2}
37 1−6iT−37T2 1 - 6iT - 37T^{2}
41 1+6T+41T2 1 + 6T + 41T^{2}
43 1−4iT−43T2 1 - 4iT - 43T^{2}
47 1+47T2 1 + 47T^{2}
53 1−2iT−53T2 1 - 2iT - 53T^{2}
59 1+4iT−59T2 1 + 4iT - 59T^{2}
61 1−2iT−61T2 1 - 2iT - 61T^{2}
67 1−4iT−67T2 1 - 4iT - 67T^{2}
71 1−8T+71T2 1 - 8T + 71T^{2}
73 1+10T+73T2 1 + 10T + 73T^{2}
79 1+8T+79T2 1 + 8T + 79T^{2}
83 1+4iT−83T2 1 + 4iT - 83T^{2}
89 1+6T+89T2 1 + 6T + 89T^{2}
97 1−2T+97T2 1 - 2T + 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

−9.293519915764770492117794630436, −8.356040830883049492056128192654, −7.938371580462713834192878948760, −6.68218875941622221512850966157, −6.42217389574043078852694510307, −5.48503749812980556220380432607, −4.37885171867923715420089821869, −3.46762236550596602135600270505, −2.71995460686128039692108259734, −1.47608256025237012465449908004, 0.32496190451294687638815284237, 1.74000517821645087039298278051, 2.69099411753690930833341423639, 4.09953842333755903220408327640, 4.63388429120213377479475858963, 5.45088594781412998577898714603, 6.39646513753830288613233671949, 7.20838689010361996648963147661, 8.094825131228622968700602592412, 8.607220065156048829403997413640

Graph of the ZZ-function along the critical line