Properties

Label 2-240-5.4-c1-0-5
Degree 22
Conductor 240240
Sign 0.447+0.894i-0.447 + 0.894i
Analytic cond. 1.916401.91640
Root an. cond. 1.384341.38434
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  i·3-s + (−2 − i)5-s − 2i·7-s − 9-s − 2·11-s − 6i·13-s + (−1 + 2i)15-s + 2i·17-s − 2·21-s − 4i·23-s + (3 + 4i)25-s + i·27-s + 8·31-s + 2i·33-s + (−2 + 4i)35-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−0.894 − 0.447i)5-s − 0.755i·7-s − 0.333·9-s − 0.603·11-s − 1.66i·13-s + (−0.258 + 0.516i)15-s + 0.485i·17-s − 0.436·21-s − 0.834i·23-s + (0.600 + 0.800i)25-s + 0.192i·27-s + 1.43·31-s + 0.348i·33-s + (−0.338 + 0.676i)35-s + ⋯

Functional equation

Λ(s)=(240s/2ΓC(s)L(s)=((0.447+0.894i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(240s/2ΓC(s+1/2)L(s)=((0.447+0.894i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 240240    =    24352^{4} \cdot 3 \cdot 5
Sign: 0.447+0.894i-0.447 + 0.894i
Analytic conductor: 1.916401.91640
Root analytic conductor: 1.384341.38434
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ240(49,)\chi_{240} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 240, ( :1/2), 0.447+0.894i)(2,\ 240,\ (\ :1/2),\ -0.447 + 0.894i)

Particular Values

L(1)L(1) \approx 0.4650770.752510i0.465077 - 0.752510i
L(12)L(\frac12) \approx 0.4650770.752510i0.465077 - 0.752510i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1+iT 1 + iT
5 1+(2+i)T 1 + (2 + i)T
good7 1+2iT7T2 1 + 2iT - 7T^{2}
11 1+2T+11T2 1 + 2T + 11T^{2}
13 1+6iT13T2 1 + 6iT - 13T^{2}
17 12iT17T2 1 - 2iT - 17T^{2}
19 1+19T2 1 + 19T^{2}
23 1+4iT23T2 1 + 4iT - 23T^{2}
29 1+29T2 1 + 29T^{2}
31 18T+31T2 1 - 8T + 31T^{2}
37 12iT37T2 1 - 2iT - 37T^{2}
41 12T+41T2 1 - 2T + 41T^{2}
43 1+4iT43T2 1 + 4iT - 43T^{2}
47 18iT47T2 1 - 8iT - 47T^{2}
53 1+6iT53T2 1 + 6iT - 53T^{2}
59 110T+59T2 1 - 10T + 59T^{2}
61 12T+61T2 1 - 2T + 61T^{2}
67 18iT67T2 1 - 8iT - 67T^{2}
71 1+12T+71T2 1 + 12T + 71T^{2}
73 14iT73T2 1 - 4iT - 73T^{2}
79 1+79T2 1 + 79T^{2}
83 1+4iT83T2 1 + 4iT - 83T^{2}
89 110T+89T2 1 - 10T + 89T^{2}
97 1+8iT97T2 1 + 8iT - 97T^{2}
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   L(s)=p j=12(1αj,pps)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

−11.97204191946259444215266705660, −10.85038935669694538881413320876, −10.13056002929891195176822211936, −8.456148144676839489131086932562, −7.957753317329978975262304578407, −7.00717100446239630133784520713, −5.61924101980997412333828064393, −4.36151973355833603134652685788, −2.98301548675506377793302167956, −0.72864190327182967399520039641, 2.58342719920451962723501294913, 3.94695391081871525948763714951, 5.02781841842898122069153631094, 6.41014651359295177422336904223, 7.53127668903399757597493950167, 8.637805473386390159780841439932, 9.499311722217538316972029861289, 10.60982430904786714834563426049, 11.65746295943836746628078584697, 11.97373279112430878810242545265

Graph of the ZZ-function along the critical line