Properties

Label 2-1176-21.17-c1-0-28
Degree 22
Conductor 11761176
Sign 0.746+0.665i-0.746 + 0.665i
Analytic cond. 9.390409.39040
Root an. cond. 3.064373.06437
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  + (−1.02 − 1.39i)3-s + (−0.244 − 0.423i)5-s + (−0.904 + 2.86i)9-s + (1.31 + 0.761i)11-s − 0.652i·13-s + (−0.341 + 0.775i)15-s + (3.77 − 6.54i)17-s + (0.364 − 0.210i)19-s + (−5.11 + 2.95i)23-s + (2.38 − 4.12i)25-s + (4.92 − 1.66i)27-s − 7.16i·29-s + (−6.39 − 3.69i)31-s + (−0.286 − 2.62i)33-s + (4.04 + 7.00i)37-s + ⋯
L(s)  = 1  + (−0.591 − 0.806i)3-s + (−0.109 − 0.189i)5-s + (−0.301 + 0.953i)9-s + (0.397 + 0.229i)11-s − 0.180i·13-s + (−0.0881 + 0.200i)15-s + (0.916 − 1.58i)17-s + (0.0836 − 0.0482i)19-s + (−1.06 + 0.616i)23-s + (0.476 − 0.824i)25-s + (0.947 − 0.320i)27-s − 1.33i·29-s + (−1.14 − 0.662i)31-s + (−0.0498 − 0.456i)33-s + (0.665 + 1.15i)37-s + ⋯

Functional equation

Λ(s)=(1176s/2ΓC(s)L(s)=((0.746+0.665i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1176s/2ΓC(s+1/2)L(s)=((0.746+0.665i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11761176    =    233722^{3} \cdot 3 \cdot 7^{2}
Sign: 0.746+0.665i-0.746 + 0.665i
Analytic conductor: 9.390409.39040
Root analytic conductor: 3.064373.06437
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1176(521,)\chi_{1176} (521, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1176, ( :1/2), 0.746+0.665i)(2,\ 1176,\ (\ :1/2),\ -0.746 + 0.665i)

Particular Values

L(1)L(1) \approx 0.91426529210.9142652921
L(12)L(\frac12) \approx 0.91426529210.9142652921
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1+(1.02+1.39i)T 1 + (1.02 + 1.39i)T
7 1 1
good5 1+(0.244+0.423i)T+(2.5+4.33i)T2 1 + (0.244 + 0.423i)T + (-2.5 + 4.33i)T^{2}
11 1+(1.310.761i)T+(5.5+9.52i)T2 1 + (-1.31 - 0.761i)T + (5.5 + 9.52i)T^{2}
13 1+0.652iT13T2 1 + 0.652iT - 13T^{2}
17 1+(3.77+6.54i)T+(8.514.7i)T2 1 + (-3.77 + 6.54i)T + (-8.5 - 14.7i)T^{2}
19 1+(0.364+0.210i)T+(9.516.4i)T2 1 + (-0.364 + 0.210i)T + (9.5 - 16.4i)T^{2}
23 1+(5.112.95i)T+(11.519.9i)T2 1 + (5.11 - 2.95i)T + (11.5 - 19.9i)T^{2}
29 1+7.16iT29T2 1 + 7.16iT - 29T^{2}
31 1+(6.39+3.69i)T+(15.5+26.8i)T2 1 + (6.39 + 3.69i)T + (15.5 + 26.8i)T^{2}
37 1+(4.047.00i)T+(18.5+32.0i)T2 1 + (-4.04 - 7.00i)T + (-18.5 + 32.0i)T^{2}
41 1+2.53T+41T2 1 + 2.53T + 41T^{2}
43 1+5.56T+43T2 1 + 5.56T + 43T^{2}
47 1+(4.65+8.06i)T+(23.5+40.7i)T2 1 + (4.65 + 8.06i)T + (-23.5 + 40.7i)T^{2}
53 1+(1.26+0.731i)T+(26.5+45.8i)T2 1 + (1.26 + 0.731i)T + (26.5 + 45.8i)T^{2}
59 1+(3.67+6.37i)T+(29.551.0i)T2 1 + (-3.67 + 6.37i)T + (-29.5 - 51.0i)T^{2}
61 1+(9.655.57i)T+(30.552.8i)T2 1 + (9.65 - 5.57i)T + (30.5 - 52.8i)T^{2}
67 1+(7.3112.6i)T+(33.558.0i)T2 1 + (7.31 - 12.6i)T + (-33.5 - 58.0i)T^{2}
71 1+12.8iT71T2 1 + 12.8iT - 71T^{2}
73 1+(8.004.62i)T+(36.5+63.2i)T2 1 + (-8.00 - 4.62i)T + (36.5 + 63.2i)T^{2}
79 1+(0.607+1.05i)T+(39.5+68.4i)T2 1 + (0.607 + 1.05i)T + (-39.5 + 68.4i)T^{2}
83 1+9.03T+83T2 1 + 9.03T + 83T^{2}
89 1+(4.77+8.27i)T+(44.5+77.0i)T2 1 + (4.77 + 8.27i)T + (-44.5 + 77.0i)T^{2}
97 14.95iT97T2 1 - 4.95iT - 97T^{2}
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   L(s)=p j=12(1αj,pps)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

−9.634726432611621213529270578886, −8.423715366853462671940624033007, −7.71643146447744692309866764238, −7.02028361401949706847650189202, −6.09809568447721236433593810332, −5.32016002597123252351415979022, −4.39854643896850132428579868633, −3.02175218455236110502640464610, −1.79047128028968487160298493196, −0.44292954919824002636692485881, 1.48997388274708261618208640141, 3.29785748944395591755135057810, 3.90551448153218601661989377158, 4.99465862450517893487424728759, 5.87060420150621987584094411771, 6.53795486884365111332551776250, 7.62824205980084766227315123141, 8.640519076375596450149843266999, 9.314120647431845621061276636893, 10.25708256349848728907225582076

Graph of the ZZ-function along the critical line