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Properties

Label 2-675-135.124-c1-0-31
Degree 22
Conductor 675675
Sign 0.961−0.275i0.961 - 0.275i
Analytic cond. 5.389905.38990
Root an. cond. 2.321612.32161
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  + (0.267 − 0.318i)2-s + (1.72 − 0.159i)3-s + (0.317 + 1.79i)4-s + (0.409 − 0.591i)6-s + (−1.29 − 0.229i)7-s + (1.37 + 0.795i)8-s + (2.94 − 0.551i)9-s + (4.90 − 1.78i)11-s + (0.834 + 3.05i)12-s + (0.0116 + 0.0138i)13-s + (−0.419 + 0.352i)14-s + (−2.81 + 1.02i)16-s + (−2.71 + 1.56i)17-s + (0.612 − 1.08i)18-s + (0.208 − 0.361i)19-s + ⋯
L(s)  = 1  + (0.188 − 0.225i)2-s + (0.995 − 0.0922i)3-s + (0.158 + 0.899i)4-s + (0.167 − 0.241i)6-s + (−0.491 − 0.0866i)7-s + (0.486 + 0.281i)8-s + (0.982 − 0.183i)9-s + (1.47 − 0.537i)11-s + (0.241 + 0.881i)12-s + (0.00321 + 0.00383i)13-s + (−0.112 + 0.0941i)14-s + (−0.703 + 0.256i)16-s + (−0.658 + 0.379i)17-s + (0.144 − 0.255i)18-s + (0.0478 − 0.0829i)19-s + ⋯

Functional equation

Λ(s)=(675s/2 ΓC(s) L(s)=((0.961−0.275i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.275i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(675s/2 ΓC(s+1/2) L(s)=((0.961−0.275i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 675675    =    33⋅523^{3} \cdot 5^{2}
Sign: 0.961−0.275i0.961 - 0.275i
Analytic conductor: 5.389905.38990
Root analytic conductor: 2.321612.32161
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ675(124,⋅)\chi_{675} (124, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 675, ( :1/2), 0.961−0.275i)(2,\ 675,\ (\ :1/2),\ 0.961 - 0.275i)

Particular Values

L(1)L(1) ≈\approx 2.45667+0.344915i2.45667 + 0.344915i
L(12)L(\frac12) ≈\approx 2.45667+0.344915i2.45667 + 0.344915i
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)
bad3 1+(−1.72+0.159i)T 1 + (-1.72 + 0.159i)T
5 1 1
good2 1+(−0.267+0.318i)T+(−0.347−1.96i)T2 1 + (-0.267 + 0.318i)T + (-0.347 - 1.96i)T^{2}
7 1+(1.29+0.229i)T+(6.57+2.39i)T2 1 + (1.29 + 0.229i)T + (6.57 + 2.39i)T^{2}
11 1+(−4.90+1.78i)T+(8.42−7.07i)T2 1 + (-4.90 + 1.78i)T + (8.42 - 7.07i)T^{2}
13 1+(−0.0116−0.0138i)T+(−2.25+12.8i)T2 1 + (-0.0116 - 0.0138i)T + (-2.25 + 12.8i)T^{2}
17 1+(2.71−1.56i)T+(8.5−14.7i)T2 1 + (2.71 - 1.56i)T + (8.5 - 14.7i)T^{2}
19 1+(−0.208+0.361i)T+(−9.5−16.4i)T2 1 + (-0.208 + 0.361i)T + (-9.5 - 16.4i)T^{2}
23 1+(1.01−0.179i)T+(21.6−7.86i)T2 1 + (1.01 - 0.179i)T + (21.6 - 7.86i)T^{2}
29 1+(−5.98−5.01i)T+(5.03+28.5i)T2 1 + (-5.98 - 5.01i)T + (5.03 + 28.5i)T^{2}
31 1+(−0.647−3.67i)T+(−29.1+10.6i)T2 1 + (-0.647 - 3.67i)T + (-29.1 + 10.6i)T^{2}
37 1+(−3.83+2.21i)T+(18.5−32.0i)T2 1 + (-3.83 + 2.21i)T + (18.5 - 32.0i)T^{2}
41 1+(2.81−2.36i)T+(7.11−40.3i)T2 1 + (2.81 - 2.36i)T + (7.11 - 40.3i)T^{2}
43 1+(2.84+7.80i)T+(−32.9+27.6i)T2 1 + (2.84 + 7.80i)T + (-32.9 + 27.6i)T^{2}
47 1+(6.99+1.23i)T+(44.1+16.0i)T2 1 + (6.99 + 1.23i)T + (44.1 + 16.0i)T^{2}
53 1−1.30iT−53T2 1 - 1.30iT - 53T^{2}
59 1+(3.47+1.26i)T+(45.1+37.9i)T2 1 + (3.47 + 1.26i)T + (45.1 + 37.9i)T^{2}
61 1+(−1.20+6.80i)T+(−57.3−20.8i)T2 1 + (-1.20 + 6.80i)T + (-57.3 - 20.8i)T^{2}
67 1+(7.08+8.44i)T+(−11.6+65.9i)T2 1 + (7.08 + 8.44i)T + (-11.6 + 65.9i)T^{2}
71 1+(−3.04−5.26i)T+(−35.5+61.4i)T2 1 + (-3.04 - 5.26i)T + (-35.5 + 61.4i)T^{2}
73 1+(0.473+0.273i)T+(36.5+63.2i)T2 1 + (0.473 + 0.273i)T + (36.5 + 63.2i)T^{2}
79 1+(0.374+0.314i)T+(13.7+77.7i)T2 1 + (0.374 + 0.314i)T + (13.7 + 77.7i)T^{2}
83 1+(−2.96+3.53i)T+(−14.4−81.7i)T2 1 + (-2.96 + 3.53i)T + (-14.4 - 81.7i)T^{2}
89 1+(1.68−2.92i)T+(−44.5−77.0i)T2 1 + (1.68 - 2.92i)T + (-44.5 - 77.0i)T^{2}
97 1+(3.40+9.34i)T+(−74.3+62.3i)T2 1 + (3.40 + 9.34i)T + (-74.3 + 62.3i)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

−10.53946569621091503188580563552, −9.452073792867935622005925072294, −8.726443631070632544078919641788, −8.144830845805705431457229666032, −6.94470916533015248601483486586, −6.49086426920036285594949847188, −4.61733593020316879967267102870, −3.68556506626407836715348994854, −3.05515276796075634440844657230, −1.69534182797887690595467793265, 1.40051767253775882849923089256, 2.63246080980123769924725029928, 4.03329416175263954167204016299, 4.75743081925946595170405204146, 6.30111969419873959695558499225, 6.71595168720651318427060114433, 7.81758307507721995164203065664, 8.937406432807379721511668068295, 9.637237737867987003908286410191, 10.07214158430828920723683593732

Graph of the ZZ-function along the critical line