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Properties

Label 2-648-9.7-c1-0-2
Degree 22
Conductor 648648
Sign −0.342−0.939i-0.342 - 0.939i
Analytic cond. 5.174305.17430
Root an. cond. 2.274712.27471
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  + (1.86 + 3.23i)5-s + (−1.73 + 3i)7-s + (1 − 1.73i)11-s + (1.23 + 2.13i)13-s − 2.26·17-s − 7.46·19-s + (−2.46 − 4.26i)23-s + (−4.46 + 7.73i)25-s + (2.13 − 3.69i)29-s + (5.46 + 9.46i)31-s − 12.9·35-s − 0.464·37-s + (3.46 + 6i)41-s + (2.26 − 3.92i)43-s + (3.46 − 6i)47-s + ⋯
L(s)  = 1  + (0.834 + 1.44i)5-s + (−0.654 + 1.13i)7-s + (0.301 − 0.522i)11-s + (0.341 + 0.591i)13-s − 0.550·17-s − 1.71·19-s + (−0.513 − 0.889i)23-s + (−0.892 + 1.54i)25-s + (0.396 − 0.686i)29-s + (0.981 + 1.69i)31-s − 2.18·35-s − 0.0762·37-s + (0.541 + 0.937i)41-s + (0.345 − 0.599i)43-s + (0.505 − 0.875i)47-s + ⋯

Functional equation

Λ(s)=(648s/2 ΓC(s) L(s)=((−0.342−0.939i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(648s/2 ΓC(s+1/2) L(s)=((−0.342−0.939i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 648648    =    23⋅342^{3} \cdot 3^{4}
Sign: −0.342−0.939i-0.342 - 0.939i
Analytic conductor: 5.174305.17430
Root analytic conductor: 2.274712.27471
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ648(217,⋅)\chi_{648} (217, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 648, ( :1/2), −0.342−0.939i)(2,\ 648,\ (\ :1/2),\ -0.342 - 0.939i)

Particular Values

L(1)L(1) ≈\approx 0.803827+1.14798i0.803827 + 1.14798i
L(12)L(\frac12) ≈\approx 0.803827+1.14798i0.803827 + 1.14798i
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+(−1.86−3.23i)T+(−2.5+4.33i)T2 1 + (-1.86 - 3.23i)T + (-2.5 + 4.33i)T^{2}
7 1+(1.73−3i)T+(−3.5−6.06i)T2 1 + (1.73 - 3i)T + (-3.5 - 6.06i)T^{2}
11 1+(−1+1.73i)T+(−5.5−9.52i)T2 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2}
13 1+(−1.23−2.13i)T+(−6.5+11.2i)T2 1 + (-1.23 - 2.13i)T + (-6.5 + 11.2i)T^{2}
17 1+2.26T+17T2 1 + 2.26T + 17T^{2}
19 1+7.46T+19T2 1 + 7.46T + 19T^{2}
23 1+(2.46+4.26i)T+(−11.5+19.9i)T2 1 + (2.46 + 4.26i)T + (-11.5 + 19.9i)T^{2}
29 1+(−2.13+3.69i)T+(−14.5−25.1i)T2 1 + (-2.13 + 3.69i)T + (-14.5 - 25.1i)T^{2}
31 1+(−5.46−9.46i)T+(−15.5+26.8i)T2 1 + (-5.46 - 9.46i)T + (-15.5 + 26.8i)T^{2}
37 1+0.464T+37T2 1 + 0.464T + 37T^{2}
41 1+(−3.46−6i)T+(−20.5+35.5i)T2 1 + (-3.46 - 6i)T + (-20.5 + 35.5i)T^{2}
43 1+(−2.26+3.92i)T+(−21.5−37.2i)T2 1 + (-2.26 + 3.92i)T + (-21.5 - 37.2i)T^{2}
47 1+(−3.46+6i)T+(−23.5−40.7i)T2 1 + (-3.46 + 6i)T + (-23.5 - 40.7i)T^{2}
53 1−10.9T+53T2 1 - 10.9T + 53T^{2}
59 1+(−4−6.92i)T+(−29.5+51.0i)T2 1 + (-4 - 6.92i)T + (-29.5 + 51.0i)T^{2}
61 1+(5.23−9.06i)T+(−30.5−52.8i)T2 1 + (5.23 - 9.06i)T + (-30.5 - 52.8i)T^{2}
67 1+(−0.267−0.464i)T+(−33.5+58.0i)T2 1 + (-0.267 - 0.464i)T + (-33.5 + 58.0i)T^{2}
71 1−2T+71T2 1 - 2T + 71T^{2}
73 1−T+73T2 1 - T + 73T^{2}
79 1+(0.267−0.464i)T+(−39.5−68.4i)T2 1 + (0.267 - 0.464i)T + (-39.5 - 68.4i)T^{2}
83 1+(1.46−2.53i)T+(−41.5−71.8i)T2 1 + (1.46 - 2.53i)T + (-41.5 - 71.8i)T^{2}
89 1−5.19T+89T2 1 - 5.19T + 89T^{2}
97 1+(−5.92+10.2i)T+(−48.5−84.0i)T2 1 + (-5.92 + 10.2i)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

−10.60077381552566648312639035669, −10.17230732993925358068589104096, −9.005695093263614897205060637678, −8.510210156719578903667854725124, −6.84365271564091683279906536926, −6.39399928466460444669654418554, −5.79018973126271885165810825743, −4.18039995764769126180089180976, −2.84266783318905781118572179705, −2.20503263524625572612342910275, 0.74472293952866474962420748327, 2.11015797825674241479677131512, 3.92859583736378663934600248803, 4.60650802936672040788348898344, 5.81792926715268055459196533350, 6.56914772412111454538092192586, 7.75564102954044241309589187220, 8.677106304019078570860908699145, 9.483037619524655563714656114900, 10.13179255602022547138628758357

Graph of the ZZ-function along the critical line