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Properties

Label 2-2520-7.2-c1-0-4
Degree 22
Conductor 25202520
Sign −0.701−0.712i-0.701 - 0.712i
Analytic cond. 20.122320.1223
Root an. cond. 4.485784.48578
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 + 2.59i)7-s + (−1 − 1.73i)11-s + 13-s + (−1.5 + 2.59i)19-s + (−0.499 − 0.866i)25-s + (4.5 + 7.79i)31-s + (−2.5 − 0.866i)35-s + (−1.5 + 2.59i)37-s − 2·41-s + 3·43-s + (−3 + 5.19i)47-s + (−6.5 + 2.59i)49-s + 1.99·55-s + (−2 − 3.46i)59-s + ⋯
L(s)  = 1  + (−0.223 + 0.387i)5-s + (0.188 + 0.981i)7-s + (−0.301 − 0.522i)11-s + 0.277·13-s + (−0.344 + 0.596i)19-s + (−0.0999 − 0.173i)25-s + (0.808 + 1.39i)31-s + (−0.422 − 0.146i)35-s + (−0.246 + 0.427i)37-s − 0.312·41-s + 0.457·43-s + (−0.437 + 0.757i)47-s + (−0.928 + 0.371i)49-s + 0.269·55-s + (−0.260 − 0.450i)59-s + ⋯

Functional equation

Λ(s)=(2520s/2 ΓC(s) L(s)=((−0.701−0.712i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2520s/2 ΓC(s+1/2) L(s)=((−0.701−0.712i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 25202520    =    23⋅32⋅5⋅72^{3} \cdot 3^{2} \cdot 5 \cdot 7
Sign: −0.701−0.712i-0.701 - 0.712i
Analytic conductor: 20.122320.1223
Root analytic conductor: 4.485784.48578
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2520(1801,⋅)\chi_{2520} (1801, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2520, ( :1/2), −0.701−0.712i)(2,\ 2520,\ (\ :1/2),\ -0.701 - 0.712i)

Particular Values

L(1)L(1) ≈\approx 1.0665166711.066516671
L(12)L(\frac12) ≈\approx 1.0665166711.066516671
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
7 1+(−0.5−2.59i)T 1 + (-0.5 - 2.59i)T
good11 1+(1+1.73i)T+(−5.5+9.52i)T2 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2}
13 1−T+13T2 1 - T + 13T^{2}
17 1+(−8.5+14.7i)T2 1 + (-8.5 + 14.7i)T^{2}
19 1+(1.5−2.59i)T+(−9.5−16.4i)T2 1 + (1.5 - 2.59i)T + (-9.5 - 16.4i)T^{2}
23 1+(−11.5−19.9i)T2 1 + (-11.5 - 19.9i)T^{2}
29 1+29T2 1 + 29T^{2}
31 1+(−4.5−7.79i)T+(−15.5+26.8i)T2 1 + (-4.5 - 7.79i)T + (-15.5 + 26.8i)T^{2}
37 1+(1.5−2.59i)T+(−18.5−32.0i)T2 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2}
41 1+2T+41T2 1 + 2T + 41T^{2}
43 1−3T+43T2 1 - 3T + 43T^{2}
47 1+(3−5.19i)T+(−23.5−40.7i)T2 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2}
53 1+(−26.5+45.8i)T2 1 + (-26.5 + 45.8i)T^{2}
59 1+(2+3.46i)T+(−29.5+51.0i)T2 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2}
61 1+(1−1.73i)T+(−30.5−52.8i)T2 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2}
67 1+(2.5+4.33i)T+(−33.5+58.0i)T2 1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2}
71 1+14T+71T2 1 + 14T + 71T^{2}
73 1+(0.5+0.866i)T+(−36.5+63.2i)T2 1 + (0.5 + 0.866i)T + (-36.5 + 63.2i)T^{2}
79 1+(4.5−7.79i)T+(−39.5−68.4i)T2 1 + (4.5 - 7.79i)T + (-39.5 - 68.4i)T^{2}
83 1−6T+83T2 1 - 6T + 83T^{2}
89 1+(2−3.46i)T+(−44.5−77.0i)T2 1 + (2 - 3.46i)T + (-44.5 - 77.0i)T^{2}
97 1+14T+97T2 1 + 14T + 97T^{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

−9.046502553664370268660308140522, −8.401340311052111018721201385101, −7.86093704983684182060545729224, −6.79008653767653431209116169550, −6.10998724273518174755242839035, −5.37012095030339318983678769393, −4.47964789811145656219081102061, −3.35496191506312556476808847210, −2.66263514703228905682009739745, −1.46578173202486610092740303495, 0.35598933204592419380338541279, 1.59851258659846327415508034260, 2.80555058472077783504835140222, 4.00574083414178767156899211747, 4.48979605561166295090958934247, 5.40591509166606710614496311835, 6.40334328231317704171973469948, 7.20967667627442631268911740403, 7.81668569693202870741659074953, 8.555803318689080057763811484465

Graph of the ZZ-function along the critical line