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Properties

Label 2-832-13.9-c1-0-16
Degree 22
Conductor 832832
Sign −0.522+0.852i-0.522 + 0.852i
Analytic cond. 6.643556.64355
Root an. cond. 2.577502.57750
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)3-s − 5-s + (−0.499 − 0.866i)9-s + (−2 + 3.46i)11-s + (−2.5 − 2.59i)13-s + (1 − 1.73i)15-s + (−1.5 − 2.59i)17-s + (1 + 1.73i)19-s + (1 − 1.73i)23-s − 4·25-s − 4.00·27-s + (2.5 − 4.33i)29-s − 2·31-s + (−3.99 − 6.92i)33-s + (2.5 − 4.33i)37-s + ⋯
L(s)  = 1  + (−0.577 + 0.999i)3-s − 0.447·5-s + (−0.166 − 0.288i)9-s + (−0.603 + 1.04i)11-s + (−0.693 − 0.720i)13-s + (0.258 − 0.447i)15-s + (−0.363 − 0.630i)17-s + (0.229 + 0.397i)19-s + (0.208 − 0.361i)23-s − 0.800·25-s − 0.769·27-s + (0.464 − 0.804i)29-s − 0.359·31-s + (−0.696 − 1.20i)33-s + (0.410 − 0.711i)37-s + ⋯

Functional equation

Λ(s)=(832s/2 ΓC(s) L(s)=((−0.522+0.852i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(832s/2 ΓC(s+1/2) L(s)=((−0.522+0.852i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 832832    =    26⋅132^{6} \cdot 13
Sign: −0.522+0.852i-0.522 + 0.852i
Analytic conductor: 6.643556.64355
Root analytic conductor: 2.577502.57750
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ832(321,⋅)\chi_{832} (321, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 11
Selberg data: (2, 832, ( :1/2), −0.522+0.852i)(2,\ 832,\ (\ :1/2),\ -0.522 + 0.852i)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
13 1+(2.5+2.59i)T 1 + (2.5 + 2.59i)T
good3 1+(1−1.73i)T+(−1.5−2.59i)T2 1 + (1 - 1.73i)T + (-1.5 - 2.59i)T^{2}
5 1+T+5T2 1 + T + 5T^{2}
7 1+(−3.5+6.06i)T2 1 + (-3.5 + 6.06i)T^{2}
11 1+(2−3.46i)T+(−5.5−9.52i)T2 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2}
17 1+(1.5+2.59i)T+(−8.5+14.7i)T2 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2}
19 1+(−1−1.73i)T+(−9.5+16.4i)T2 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2}
23 1+(−1+1.73i)T+(−11.5−19.9i)T2 1 + (-1 + 1.73i)T + (-11.5 - 19.9i)T^{2}
29 1+(−2.5+4.33i)T+(−14.5−25.1i)T2 1 + (-2.5 + 4.33i)T + (-14.5 - 25.1i)T^{2}
31 1+2T+31T2 1 + 2T + 31T^{2}
37 1+(−2.5+4.33i)T+(−18.5−32.0i)T2 1 + (-2.5 + 4.33i)T + (-18.5 - 32.0i)T^{2}
41 1+(1.5−2.59i)T+(−20.5−35.5i)T2 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2}
43 1+(−2−3.46i)T+(−21.5+37.2i)T2 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2}
47 1+6T+47T2 1 + 6T + 47T^{2}
53 1+13T+53T2 1 + 13T + 53T^{2}
59 1+(6+10.3i)T+(−29.5+51.0i)T2 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2}
61 1+(3.5+6.06i)T+(−30.5+52.8i)T2 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2}
67 1+(−7+12.1i)T+(−33.5−58.0i)T2 1 + (-7 + 12.1i)T + (-33.5 - 58.0i)T^{2}
71 1+(−3−5.19i)T+(−35.5+61.4i)T2 1 + (-3 - 5.19i)T + (-35.5 + 61.4i)T^{2}
73 1−7T+73T2 1 - 7T + 73T^{2}
79 1+8T+79T2 1 + 8T + 79T^{2}
83 1−4T+83T2 1 - 4T + 83T^{2}
89 1+(7−12.1i)T+(−44.5−77.0i)T2 1 + (7 - 12.1i)T + (-44.5 - 77.0i)T^{2}
97 1+(−1−1.73i)T+(−48.5+84.0i)T2 1 + (-1 - 1.73i)T + (-48.5 + 84.0i)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

−9.755939481145026765633752700028, −9.630209143603333530824364613625, −8.034811696720701676218661773854, −7.54637921581084797412542331439, −6.33611272842059421796395011230, −5.11622260764358190081503364725, −4.74305986137614045969954477516, −3.65562182532463787298423412756, −2.30922507337412025737703628465, 0, 1.50748275186166494399882492118, 2.93186251521172723495321887796, 4.21262761918945774011885828446, 5.40004752762507478010660547365, 6.24332479928020471850298565135, 7.07370601698036457028517533921, 7.77262418328361040576312245365, 8.660443607351934605163449136980, 9.629696918347909247258172870597

Graph of the ZZ-function along the critical line