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Properties

Label 2-3240-9.7-c1-0-6
Degree 22
Conductor 32403240
Sign 0.173−0.984i0.173 - 0.984i
Analytic cond. 25.871525.8715
Root an. cond. 5.086405.08640
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  + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)11-s − 7·19-s + (3 + 5.19i)23-s + (−0.499 + 0.866i)25-s + (−3.5 + 6.06i)29-s + (−0.5 − 0.866i)31-s − 2·37-s + (4.5 + 7.79i)41-s + (3 − 5.19i)43-s + (−1 + 1.73i)47-s + (3.5 + 6.06i)49-s − 0.999·55-s + (1.5 + 2.59i)59-s + (5 − 8.66i)61-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (0.150 − 0.261i)11-s − 1.60·19-s + (0.625 + 1.08i)23-s + (−0.0999 + 0.173i)25-s + (−0.649 + 1.12i)29-s + (−0.0898 − 0.155i)31-s − 0.328·37-s + (0.702 + 1.21i)41-s + (0.457 − 0.792i)43-s + (−0.145 + 0.252i)47-s + (0.5 + 0.866i)49-s − 0.134·55-s + (0.195 + 0.338i)59-s + (0.640 − 1.10i)61-s + ⋯

Functional equation

Λ(s)=(3240s/2 ΓC(s) L(s)=((0.173−0.984i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3240s/2 ΓC(s+1/2) L(s)=((0.173−0.984i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32403240    =    23⋅34⋅52^{3} \cdot 3^{4} \cdot 5
Sign: 0.173−0.984i0.173 - 0.984i
Analytic conductor: 25.871525.8715
Root analytic conductor: 5.086405.08640
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3240(2161,⋅)\chi_{3240} (2161, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3240, ( :1/2), 0.173−0.984i)(2,\ 3240,\ (\ :1/2),\ 0.173 - 0.984i)

Particular Values

L(1)L(1) ≈\approx 1.1218558371.121855837
L(12)L(\frac12) ≈\approx 1.1218558371.121855837
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
5 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
good7 1+(−3.5−6.06i)T2 1 + (-3.5 - 6.06i)T^{2}
11 1+(−0.5+0.866i)T+(−5.5−9.52i)T2 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2}
13 1+(−6.5+11.2i)T2 1 + (-6.5 + 11.2i)T^{2}
17 1+17T2 1 + 17T^{2}
19 1+7T+19T2 1 + 7T + 19T^{2}
23 1+(−3−5.19i)T+(−11.5+19.9i)T2 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2}
29 1+(3.5−6.06i)T+(−14.5−25.1i)T2 1 + (3.5 - 6.06i)T + (-14.5 - 25.1i)T^{2}
31 1+(0.5+0.866i)T+(−15.5+26.8i)T2 1 + (0.5 + 0.866i)T + (-15.5 + 26.8i)T^{2}
37 1+2T+37T2 1 + 2T + 37T^{2}
41 1+(−4.5−7.79i)T+(−20.5+35.5i)T2 1 + (-4.5 - 7.79i)T + (-20.5 + 35.5i)T^{2}
43 1+(−3+5.19i)T+(−21.5−37.2i)T2 1 + (-3 + 5.19i)T + (-21.5 - 37.2i)T^{2}
47 1+(1−1.73i)T+(−23.5−40.7i)T2 1 + (1 - 1.73i)T + (-23.5 - 40.7i)T^{2}
53 1+53T2 1 + 53T^{2}
59 1+(−1.5−2.59i)T+(−29.5+51.0i)T2 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2}
61 1+(−5+8.66i)T+(−30.5−52.8i)T2 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2}
67 1+(−1−1.73i)T+(−33.5+58.0i)T2 1 + (-1 - 1.73i)T + (-33.5 + 58.0i)T^{2}
71 1+T+71T2 1 + T + 71T^{2}
73 1+73T2 1 + 73T^{2}
79 1+(2−3.46i)T+(−39.5−68.4i)T2 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2}
83 1+(3−5.19i)T+(−41.5−71.8i)T2 1 + (3 - 5.19i)T + (-41.5 - 71.8i)T^{2}
89 1−7T+89T2 1 - 7T + 89T^{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

−8.886969200731195783610123488199, −8.110719199251530693463066107274, −7.37386854581611387259722473585, −6.58573833081599472373357771697, −5.77059179378850071528799696860, −4.98969988818598053042667425669, −4.13311091801798342121841708923, −3.37042191233296151122071031471, −2.22644791967771007060631995971, −1.11717829738032397418185346735, 0.37161786521798621901467683054, 1.95948385673889945936172138544, 2.73753060388801700091898098854, 3.91854136012179800051379803532, 4.43388319362375098944391287347, 5.50778085643179359860210389790, 6.34748410794748119229715275041, 6.94626203376860517157058496202, 7.72107542061064072730207554392, 8.546427124431478521912065956665

Graph of the ZZ-function along the critical line