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Properties

Label 2-2548-13.3-c1-0-20
Degree 22
Conductor 25482548
Sign 0.522+0.852i0.522 + 0.852i
Analytic cond. 20.345820.3458
Root an. cond. 4.510644.51064
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  − 3·5-s + (1.5 − 2.59i)9-s + (1 + 1.73i)11-s + (2.5 − 2.59i)13-s + (−3.5 + 6.06i)17-s + (1 − 1.73i)19-s + (2 + 3.46i)23-s + 4·25-s + (−0.5 − 0.866i)29-s − 4·31-s + (−0.5 − 0.866i)37-s + (−1.5 − 2.59i)41-s + (3 − 5.19i)43-s + (−4.5 + 7.79i)45-s + 10·47-s + ⋯
L(s)  = 1  − 1.34·5-s + (0.5 − 0.866i)9-s + (0.301 + 0.522i)11-s + (0.693 − 0.720i)13-s + (−0.848 + 1.47i)17-s + (0.229 − 0.397i)19-s + (0.417 + 0.722i)23-s + 0.800·25-s + (−0.0928 − 0.160i)29-s − 0.718·31-s + (−0.0821 − 0.142i)37-s + (−0.234 − 0.405i)41-s + (0.457 − 0.792i)43-s + (−0.670 + 1.16i)45-s + 1.45·47-s + ⋯

Functional equation

Λ(s)=(2548s/2 ΓC(s) L(s)=((0.522+0.852i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2548s/2 ΓC(s+1/2) L(s)=((0.522+0.852i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{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: 25482548    =    22⋅72⋅132^{2} \cdot 7^{2} \cdot 13
Sign: 0.522+0.852i0.522 + 0.852i
Analytic conductor: 20.345820.3458
Root analytic conductor: 4.510644.51064
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2548(393,⋅)\chi_{2548} (393, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2548, ( :1/2), 0.522+0.852i)(2,\ 2548,\ (\ :1/2),\ 0.522 + 0.852i)

Particular Values

L(1)L(1) ≈\approx 1.2124654371.212465437
L(12)L(\frac12) ≈\approx 1.2124654371.212465437
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
7 1 1
13 1+(−2.5+2.59i)T 1 + (-2.5 + 2.59i)T
good3 1+(−1.5+2.59i)T2 1 + (-1.5 + 2.59i)T^{2}
5 1+3T+5T2 1 + 3T + 5T^{2}
11 1+(−1−1.73i)T+(−5.5+9.52i)T2 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2}
17 1+(3.5−6.06i)T+(−8.5−14.7i)T2 1 + (3.5 - 6.06i)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+(−2−3.46i)T+(−11.5+19.9i)T2 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2}
29 1+(0.5+0.866i)T+(−14.5+25.1i)T2 1 + (0.5 + 0.866i)T + (-14.5 + 25.1i)T^{2}
31 1+4T+31T2 1 + 4T + 31T^{2}
37 1+(0.5+0.866i)T+(−18.5+32.0i)T2 1 + (0.5 + 0.866i)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+(−3+5.19i)T+(−21.5−37.2i)T2 1 + (-3 + 5.19i)T + (-21.5 - 37.2i)T^{2}
47 1−10T+47T2 1 - 10T + 47T^{2}
53 1+7T+53T2 1 + 7T + 53T^{2}
59 1+(−3+5.19i)T+(−29.5−51.0i)T2 1 + (-3 + 5.19i)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+(4+6.92i)T+(−33.5+58.0i)T2 1 + (4 + 6.92i)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 1−11T+73T2 1 - 11T + 73T^{2}
79 1+14T+79T2 1 + 14T + 79T^{2}
83 1−14T+83T2 1 - 14T + 83T^{2}
89 1+(5+8.66i)T+(−44.5+77.0i)T2 1 + (5 + 8.66i)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

−8.771533196099515226546872989751, −7.963161202791244164806841281602, −7.29976089567537064456536202350, −6.60547740762340882038418690499, −5.73652921882286817945591531776, −4.58671649043391313875551568453, −3.80506928903124666121220395684, −3.42700793818715616641761581894, −1.82193537847506765450927324581, −0.52528546197402985772811021165, 0.960025583495519507257773129472, 2.37465898505501122014658200883, 3.46287247027335276241163590520, 4.28696826401480677940843569730, 4.83395384720296235533461168517, 5.96888221250178225841222573394, 7.04847782351937394323935321024, 7.35503276788284135711758507685, 8.318224048353224040721873255330, 8.834407231177779058629724143036

Graph of the ZZ-function along the critical line