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Properties

Label 2-3240-9.4-c1-0-25
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 + (−2.13 − 3.70i)7-s + (0.637 + 1.10i)11-s + (−3.13 + 5.43i)13-s + 2·17-s + 19-s + (−0.137 + 0.238i)23-s + (−0.499 − 0.866i)25-s + (0.637 + 1.10i)29-s + (0.637 − 1.10i)31-s + 4.27·35-s + 4.54·37-s + (3.77 − 6.53i)41-s + (−2 − 3.46i)43-s + (−3.13 − 5.43i)47-s + ⋯
L(s)  = 1  + (−0.223 + 0.387i)5-s + (−0.807 − 1.39i)7-s + (0.192 + 0.332i)11-s + (−0.870 + 1.50i)13-s + 0.485·17-s + 0.229·19-s + (−0.0286 + 0.0496i)23-s + (−0.0999 − 0.173i)25-s + (0.118 + 0.205i)29-s + (0.114 − 0.198i)31-s + 0.722·35-s + 0.747·37-s + (0.589 − 1.02i)41-s + (−0.304 − 0.528i)43-s + (−0.457 − 0.792i)47-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(1081,⋅)\chi_{3240} (1081, \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 1.0748419231.074841923
L(12)L(\frac12) ≈\approx 1.0748419231.074841923
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+(2.13+3.70i)T+(−3.5+6.06i)T2 1 + (2.13 + 3.70i)T + (-3.5 + 6.06i)T^{2}
11 1+(−0.637−1.10i)T+(−5.5+9.52i)T2 1 + (-0.637 - 1.10i)T + (-5.5 + 9.52i)T^{2}
13 1+(3.13−5.43i)T+(−6.5−11.2i)T2 1 + (3.13 - 5.43i)T + (-6.5 - 11.2i)T^{2}
17 1−2T+17T2 1 - 2T + 17T^{2}
19 1−T+19T2 1 - T + 19T^{2}
23 1+(0.137−0.238i)T+(−11.5−19.9i)T2 1 + (0.137 - 0.238i)T + (-11.5 - 19.9i)T^{2}
29 1+(−0.637−1.10i)T+(−14.5+25.1i)T2 1 + (-0.637 - 1.10i)T + (-14.5 + 25.1i)T^{2}
31 1+(−0.637+1.10i)T+(−15.5−26.8i)T2 1 + (-0.637 + 1.10i)T + (-15.5 - 26.8i)T^{2}
37 1−4.54T+37T2 1 - 4.54T + 37T^{2}
41 1+(−3.77+6.53i)T+(−20.5−35.5i)T2 1 + (-3.77 + 6.53i)T + (-20.5 - 35.5i)T^{2}
43 1+(2+3.46i)T+(−21.5+37.2i)T2 1 + (2 + 3.46i)T + (-21.5 + 37.2i)T^{2}
47 1+(3.13+5.43i)T+(−23.5+40.7i)T2 1 + (3.13 + 5.43i)T + (-23.5 + 40.7i)T^{2}
53 1−8.27T+53T2 1 - 8.27T + 53T^{2}
59 1+(−6.5+11.2i)T+(−29.5−51.0i)T2 1 + (-6.5 + 11.2i)T + (-29.5 - 51.0i)T^{2}
61 1+(−3.27−5.67i)T+(−30.5+52.8i)T2 1 + (-3.27 - 5.67i)T + (-30.5 + 52.8i)T^{2}
67 1+(−7.27+12.6i)T+(−33.5−58.0i)T2 1 + (-7.27 + 12.6i)T + (-33.5 - 58.0i)T^{2}
71 1+0.725T+71T2 1 + 0.725T + 71T^{2}
73 1+15.0T+73T2 1 + 15.0T + 73T^{2}
79 1+(−2.27−3.94i)T+(−39.5+68.4i)T2 1 + (-2.27 - 3.94i)T + (-39.5 + 68.4i)T^{2}
83 1+(6.27+10.8i)T+(−41.5+71.8i)T2 1 + (6.27 + 10.8i)T + (-41.5 + 71.8i)T^{2}
89 1+9.82T+89T2 1 + 9.82T + 89T^{2}
97 1+(8+13.8i)T+(−48.5+84.0i)T2 1 + (8 + 13.8i)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

−8.455807225540462834214002409533, −7.42978968680404756991616881469, −7.03836172448920726817286488711, −6.56232871458853673207573016180, −5.43590690678611229897868269154, −4.31591032022742999178370831275, −3.93400762030274299023364035412, −2.93731378285775187473723937460, −1.79057216632657186770698716915, −0.39213207595486492653136790845, 0.988441127705432430257004561055, 2.63174143404559382185289653467, 2.96663573115802318698406795440, 4.13549531751586077961136025781, 5.28641907115237213210037992604, 5.63661864379034776542001094716, 6.43656546525843915890479731140, 7.44465769534680243588252108954, 8.199552401943200579518984165449, 8.716584123807434904046316709762

Graph of the ZZ-function along the critical line