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Properties

Label 2-360-9.4-c1-0-7
Degree 22
Conductor 360360
Sign 0.986+0.164i0.986 + 0.164i
Analytic cond. 2.874612.87461
Root an. cond. 1.695461.69546
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  + (1.72 − 0.158i)3-s + (0.5 − 0.866i)5-s + (0.724 + 1.25i)7-s + (2.94 − 0.548i)9-s + (0.724 − 1.57i)15-s − 2·17-s + 2.89·19-s + (1.44 + 2.04i)21-s + (1.27 − 2.20i)23-s + (−0.499 − 0.866i)25-s + (4.99 − 1.41i)27-s + (−3.94 − 6.84i)29-s + (−5.44 + 9.43i)31-s + 1.44·35-s − 6·37-s + ⋯
L(s)  = 1  + (0.995 − 0.0917i)3-s + (0.223 − 0.387i)5-s + (0.273 + 0.474i)7-s + (0.983 − 0.182i)9-s + (0.187 − 0.406i)15-s − 0.485·17-s + 0.665·19-s + (0.316 + 0.447i)21-s + (0.265 − 0.460i)23-s + (−0.0999 − 0.173i)25-s + (0.962 − 0.272i)27-s + (−0.733 − 1.27i)29-s + (−0.978 + 1.69i)31-s + 0.245·35-s − 0.986·37-s + ⋯

Functional equation

Λ(s)=(360s/2 ΓC(s) L(s)=((0.986+0.164i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.164i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(360s/2 ΓC(s+1/2) L(s)=((0.986+0.164i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 360360    =    23⋅32⋅52^{3} \cdot 3^{2} \cdot 5
Sign: 0.986+0.164i0.986 + 0.164i
Analytic conductor: 2.874612.87461
Root analytic conductor: 1.695461.69546
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ360(121,⋅)\chi_{360} (121, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 360, ( :1/2), 0.986+0.164i)(2,\ 360,\ (\ :1/2),\ 0.986 + 0.164i)

Particular Values

L(1)L(1) ≈\approx 1.94747−0.161330i1.94747 - 0.161330i
L(12)L(\frac12) ≈\approx 1.94747−0.161330i1.94747 - 0.161330i
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+(−1.72+0.158i)T 1 + (-1.72 + 0.158i)T
5 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
good7 1+(−0.724−1.25i)T+(−3.5+6.06i)T2 1 + (-0.724 - 1.25i)T + (-3.5 + 6.06i)T^{2}
11 1+(−5.5+9.52i)T2 1 + (-5.5 + 9.52i)T^{2}
13 1+(−6.5−11.2i)T2 1 + (-6.5 - 11.2i)T^{2}
17 1+2T+17T2 1 + 2T + 17T^{2}
19 1−2.89T+19T2 1 - 2.89T + 19T^{2}
23 1+(−1.27+2.20i)T+(−11.5−19.9i)T2 1 + (-1.27 + 2.20i)T + (-11.5 - 19.9i)T^{2}
29 1+(3.94+6.84i)T+(−14.5+25.1i)T2 1 + (3.94 + 6.84i)T + (-14.5 + 25.1i)T^{2}
31 1+(5.44−9.43i)T+(−15.5−26.8i)T2 1 + (5.44 - 9.43i)T + (-15.5 - 26.8i)T^{2}
37 1+6T+37T2 1 + 6T + 37T^{2}
41 1+(−0.0505+0.0874i)T+(−20.5−35.5i)T2 1 + (-0.0505 + 0.0874i)T + (-20.5 - 35.5i)T^{2}
43 1+(−3.89−6.75i)T+(−21.5+37.2i)T2 1 + (-3.89 - 6.75i)T + (-21.5 + 37.2i)T^{2}
47 1+(−2.27−3.94i)T+(−23.5+40.7i)T2 1 + (-2.27 - 3.94i)T + (-23.5 + 40.7i)T^{2}
53 1+11.7T+53T2 1 + 11.7T + 53T^{2}
59 1+(−5.44+9.43i)T+(−29.5−51.0i)T2 1 + (-5.44 + 9.43i)T + (-29.5 - 51.0i)T^{2}
61 1+(−1.5−2.59i)T+(−30.5+52.8i)T2 1 + (-1.5 - 2.59i)T + (-30.5 + 52.8i)T^{2}
67 1+(5.62−9.74i)T+(−33.5−58.0i)T2 1 + (5.62 - 9.74i)T + (-33.5 - 58.0i)T^{2}
71 1+9.79T+71T2 1 + 9.79T + 71T^{2}
73 1+5.79T+73T2 1 + 5.79T + 73T^{2}
79 1+(1.44+2.51i)T+(−39.5+68.4i)T2 1 + (1.44 + 2.51i)T + (-39.5 + 68.4i)T^{2}
83 1+(−0.275−0.476i)T+(−41.5+71.8i)T2 1 + (-0.275 - 0.476i)T + (-41.5 + 71.8i)T^{2}
89 1+16.7T+89T2 1 + 16.7T + 89T^{2}
97 1+(1+1.73i)T+(−48.5+84.0i)T2 1 + (1 + 1.73i)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

−11.48030508330562746998505522395, −10.33025784353295685360432515962, −9.330842616536533941992812676537, −8.737107781005639933990326748194, −7.81992017963868845293862917471, −6.80474761404584793533608593350, −5.46436336555606318863860802678, −4.31804690430714939877816087568, −2.98189148802267898869008997397, −1.69963129427529424521091621478, 1.80144596751132884212957624817, 3.18251269739172063177625062524, 4.21677105280676442361984230932, 5.56119023600477559060575936366, 7.07107515254452554170514653658, 7.58033622860044370313005924892, 8.803629329622636128334258934669, 9.511493626184509553261518477049, 10.49969951836145950144410732875, 11.26024076594844072621895541245

Graph of the ZZ-function along the critical line