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Properties

Label 2-245-7.2-c3-0-32
Degree 22
Conductor 245245
Sign −0.605+0.795i-0.605 + 0.795i
Analytic cond. 14.455414.4554
Root an. cond. 3.802033.80203
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 + 2.59i)2-s + (−1 − 1.73i)3-s + (−0.5 − 0.866i)4-s + (2.5 − 4.33i)5-s + 6·6-s − 21·8-s + (11.5 − 19.9i)9-s + (7.50 + 12.9i)10-s + (22.5 + 38.9i)11-s + (−1.00 + 1.73i)12-s − 59·13-s − 10·15-s + (35.5 − 61.4i)16-s + (−27 − 46.7i)17-s + (34.5 + 59.7i)18-s + (−60.5 + 104. i)19-s + ⋯
L(s)  = 1  + (−0.530 + 0.918i)2-s + (−0.192 − 0.333i)3-s + (−0.0625 − 0.108i)4-s + (0.223 − 0.387i)5-s + 0.408·6-s − 0.928·8-s + (0.425 − 0.737i)9-s + (0.237 + 0.410i)10-s + (0.616 + 1.06i)11-s + (−0.0240 + 0.0416i)12-s − 1.25·13-s − 0.172·15-s + (0.554 − 0.960i)16-s + (−0.385 − 0.667i)17-s + (0.451 + 0.782i)18-s + (−0.730 + 1.26i)19-s + ⋯

Functional equation

Λ(s)=(245s/2 ΓC(s) L(s)=((−0.605+0.795i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(245s/2 ΓC(s+3/2) L(s)=((−0.605+0.795i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 245245    =    5⋅725 \cdot 7^{2}
Sign: −0.605+0.795i-0.605 + 0.795i
Analytic conductor: 14.455414.4554
Root analytic conductor: 3.802033.80203
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ245(226,⋅)\chi_{245} (226, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 11
Selberg data: (2, 245, ( :3/2), −0.605+0.795i)(2,\ 245,\ (\ :3/2),\ -0.605 + 0.795i)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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)
bad5 1+(−2.5+4.33i)T 1 + (-2.5 + 4.33i)T
7 1 1
good2 1+(1.5−2.59i)T+(−4−6.92i)T2 1 + (1.5 - 2.59i)T + (-4 - 6.92i)T^{2}
3 1+(1+1.73i)T+(−13.5+23.3i)T2 1 + (1 + 1.73i)T + (-13.5 + 23.3i)T^{2}
11 1+(−22.5−38.9i)T+(−665.5+1.15e3i)T2 1 + (-22.5 - 38.9i)T + (-665.5 + 1.15e3i)T^{2}
13 1+59T+2.19e3T2 1 + 59T + 2.19e3T^{2}
17 1+(27+46.7i)T+(−2.45e3+4.25e3i)T2 1 + (27 + 46.7i)T + (-2.45e3 + 4.25e3i)T^{2}
19 1+(60.5−104.i)T+(−3.42e3−5.94e3i)T2 1 + (60.5 - 104. i)T + (-3.42e3 - 5.94e3i)T^{2}
23 1+(34.5−59.7i)T+(−6.08e3−1.05e4i)T2 1 + (34.5 - 59.7i)T + (-6.08e3 - 1.05e4i)T^{2}
29 1+162T+2.43e4T2 1 + 162T + 2.43e4T^{2}
31 1+(44+76.2i)T+(−1.48e4+2.57e4i)T2 1 + (44 + 76.2i)T + (-1.48e4 + 2.57e4i)T^{2}
37 1+(−129.5+224.i)T+(−2.53e4−4.38e4i)T2 1 + (-129.5 + 224. i)T + (-2.53e4 - 4.38e4i)T^{2}
41 1+195T+6.89e4T2 1 + 195T + 6.89e4T^{2}
43 1+286T+7.95e4T2 1 + 286T + 7.95e4T^{2}
47 1+(−22.5+38.9i)T+(−5.19e4−8.99e4i)T2 1 + (-22.5 + 38.9i)T + (-5.19e4 - 8.99e4i)T^{2}
53 1+(298.5+517.i)T+(−7.44e4+1.28e5i)T2 1 + (298.5 + 517. i)T + (-7.44e4 + 1.28e5i)T^{2}
59 1+(180+311.i)T+(−1.02e5+1.77e5i)T2 1 + (180 + 311. i)T + (-1.02e5 + 1.77e5i)T^{2}
61 1+(−196+339.i)T+(−1.13e5−1.96e5i)T2 1 + (-196 + 339. i)T + (-1.13e5 - 1.96e5i)T^{2}
67 1+(−140−242.i)T+(−1.50e5+2.60e5i)T2 1 + (-140 - 242. i)T + (-1.50e5 + 2.60e5i)T^{2}
71 1−48T+3.57e5T2 1 - 48T + 3.57e5T^{2}
73 1+(−334−578.i)T+(−1.94e5+3.36e5i)T2 1 + (-334 - 578. i)T + (-1.94e5 + 3.36e5i)T^{2}
79 1+(391−677.i)T+(−2.46e5−4.26e5i)T2 1 + (391 - 677. i)T + (-2.46e5 - 4.26e5i)T^{2}
83 1+768T+5.71e5T2 1 + 768T + 5.71e5T^{2}
89 1+(597−1.03e3i)T+(−3.52e5−6.10e5i)T2 1 + (597 - 1.03e3i)T + (-3.52e5 - 6.10e5i)T^{2}
97 1+902T+9.12e5T2 1 + 902T + 9.12e5T^{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

−11.58267209736287399651128661448, −9.672811118446763062863025039888, −9.542129607324851715494398006041, −8.148237290720854196389580006949, −7.21325014567227958757266482476, −6.58319668320953836951187300449, −5.39143264305443359615586135431, −3.92838727740324194656104250926, −1.95687392609680384929042194330, 0, 1.80984154891676207836866886978, 2.97775683614366429770859189630, 4.52165325647870607520955010196, 5.88957691589284154573896068572, 6.95935213476336409337267378419, 8.428577478688992043387629518628, 9.381113582207653616508340074642, 10.26021769626393133744860111608, 10.92346992168105237331954304574

Graph of the ZZ-function along the critical line