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Properties

Label 2-2592-9.7-c1-0-29
Degree 22
Conductor 25922592
Sign 0.766+0.642i0.766 + 0.642i
Analytic cond. 20.697220.6972
Root an. cond. 4.549424.54942
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 + 1.73i)5-s + (−3 − 5.19i)13-s + 2·17-s + (0.500 − 0.866i)25-s + (5 − 8.66i)29-s − 2·37-s + (−5 − 8.66i)41-s + (3.5 + 6.06i)49-s + 14·53-s + (5 − 8.66i)61-s + (6 − 10.3i)65-s − 6·73-s + (2 + 3.46i)85-s + 10·89-s + (−9 + 15.5i)97-s + ⋯
L(s)  = 1  + (0.447 + 0.774i)5-s + (−0.832 − 1.44i)13-s + 0.485·17-s + (0.100 − 0.173i)25-s + (0.928 − 1.60i)29-s − 0.328·37-s + (−0.780 − 1.35i)41-s + (0.5 + 0.866i)49-s + 1.92·53-s + (0.640 − 1.10i)61-s + (0.744 − 1.28i)65-s − 0.702·73-s + (0.216 + 0.375i)85-s + 1.05·89-s + (−0.913 + 1.58i)97-s + ⋯

Functional equation

Λ(s)=(2592s/2 ΓC(s) L(s)=((0.766+0.642i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2592s/2 ΓC(s+1/2) L(s)=((0.766+0.642i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 25922592    =    25⋅342^{5} \cdot 3^{4}
Sign: 0.766+0.642i0.766 + 0.642i
Analytic conductor: 20.697220.6972
Root analytic conductor: 4.549424.54942
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2592(865,⋅)\chi_{2592} (865, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2592, ( :1/2), 0.766+0.642i)(2,\ 2592,\ (\ :1/2),\ 0.766 + 0.642i)

Particular Values

L(1)L(1) ≈\approx 1.7480383691.748038369
L(12)L(\frac12) ≈\approx 1.7480383691.748038369
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+(−1−1.73i)T+(−2.5+4.33i)T2 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2}
7 1+(−3.5−6.06i)T2 1 + (-3.5 - 6.06i)T^{2}
11 1+(−5.5−9.52i)T2 1 + (-5.5 - 9.52i)T^{2}
13 1+(3+5.19i)T+(−6.5+11.2i)T2 1 + (3 + 5.19i)T + (-6.5 + 11.2i)T^{2}
17 1−2T+17T2 1 - 2T + 17T^{2}
19 1+19T2 1 + 19T^{2}
23 1+(−11.5+19.9i)T2 1 + (-11.5 + 19.9i)T^{2}
29 1+(−5+8.66i)T+(−14.5−25.1i)T2 1 + (-5 + 8.66i)T + (-14.5 - 25.1i)T^{2}
31 1+(−15.5+26.8i)T2 1 + (-15.5 + 26.8i)T^{2}
37 1+2T+37T2 1 + 2T + 37T^{2}
41 1+(5+8.66i)T+(−20.5+35.5i)T2 1 + (5 + 8.66i)T + (-20.5 + 35.5i)T^{2}
43 1+(−21.5−37.2i)T2 1 + (-21.5 - 37.2i)T^{2}
47 1+(−23.5−40.7i)T2 1 + (-23.5 - 40.7i)T^{2}
53 1−14T+53T2 1 - 14T + 53T^{2}
59 1+(−29.5+51.0i)T2 1 + (-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+(−33.5+58.0i)T2 1 + (-33.5 + 58.0i)T^{2}
71 1+71T2 1 + 71T^{2}
73 1+6T+73T2 1 + 6T + 73T^{2}
79 1+(−39.5−68.4i)T2 1 + (-39.5 - 68.4i)T^{2}
83 1+(−41.5−71.8i)T2 1 + (-41.5 - 71.8i)T^{2}
89 1−10T+89T2 1 - 10T + 89T^{2}
97 1+(9−15.5i)T+(−48.5−84.0i)T2 1 + (9 - 15.5i)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

−8.744459646820643702449843284656, −7.941724966536288114315934022714, −7.31353685290245761884017379474, −6.47896843890878725352938298492, −5.71103980586354661400137151270, −5.02106346293686316784573436189, −3.87641358932578115040214780271, −2.88575589535561264730649272834, −2.26182886860895408517592546414, −0.63251589691096042975129502053, 1.16144321176864410002697463006, 2.09528771784926336716786789950, 3.24902404234064968970224200129, 4.39220499847601658592016426977, 5.00184198945169374115743981709, 5.74753818253531788409337612762, 6.83690249519887679754324698026, 7.24573841553135259337695054122, 8.520443249114953747784839538589, 8.834070438261762304652540793764

Graph of the ZZ-function along the critical line