Properties

Label 2-55-11.10-c2-0-4
Degree 22
Conductor 5555
Sign 0.535+0.844i0.535 + 0.844i
Analytic cond. 1.498641.49864
Root an. cond. 1.224191.22419
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.46i·2-s + 0.114·3-s + 1.86·4-s + 2.23·5-s − 0.167i·6-s − 4.56i·7-s − 8.56i·8-s − 8.98·9-s − 3.26i·10-s + (5.89 + 9.28i)11-s + 0.213·12-s + 16.5i·13-s − 6.66·14-s + 0.256·15-s − 5.05·16-s − 17.2i·17-s + ⋯
L(s)  = 1  − 0.730i·2-s + 0.0381·3-s + 0.466·4-s + 0.447·5-s − 0.0278i·6-s − 0.651i·7-s − 1.07i·8-s − 0.998·9-s − 0.326i·10-s + (0.535 + 0.844i)11-s + 0.0178·12-s + 1.27i·13-s − 0.475·14-s + 0.0170·15-s − 0.316·16-s − 1.01i·17-s + ⋯

Functional equation

Λ(s)=(55s/2ΓC(s)L(s)=((0.535+0.844i)Λ(3s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(55s/2ΓC(s+1)L(s)=((0.535+0.844i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5555    =    5115 \cdot 11
Sign: 0.535+0.844i0.535 + 0.844i
Analytic conductor: 1.498641.49864
Root analytic conductor: 1.224191.22419
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ55(21,)\chi_{55} (21, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 55, ( :1), 0.535+0.844i)(2,\ 55,\ (\ :1),\ 0.535 + 0.844i)

Particular Values

L(32)L(\frac{3}{2}) \approx 1.173950.645527i1.17395 - 0.645527i
L(12)L(\frac12) \approx 1.173950.645527i1.17395 - 0.645527i
L(2)L(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)
bad5 12.23T 1 - 2.23T
11 1+(5.899.28i)T 1 + (-5.89 - 9.28i)T
good2 1+1.46iT4T2 1 + 1.46iT - 4T^{2}
3 10.114T+9T2 1 - 0.114T + 9T^{2}
7 1+4.56iT49T2 1 + 4.56iT - 49T^{2}
13 116.5iT169T2 1 - 16.5iT - 169T^{2}
17 1+17.2iT289T2 1 + 17.2iT - 289T^{2}
19 135.8iT361T2 1 - 35.8iT - 361T^{2}
23 1+29.3T+529T2 1 + 29.3T + 529T^{2}
29 1+8.51iT841T2 1 + 8.51iT - 841T^{2}
31 1+26.3T+961T2 1 + 26.3T + 961T^{2}
37 144.4T+1.36e3T2 1 - 44.4T + 1.36e3T^{2}
41 1+52.2iT1.68e3T2 1 + 52.2iT - 1.68e3T^{2}
43 1+6.77iT1.84e3T2 1 + 6.77iT - 1.84e3T^{2}
47 115.0T+2.20e3T2 1 - 15.0T + 2.20e3T^{2}
53 1+33.1T+2.80e3T2 1 + 33.1T + 2.80e3T^{2}
59 151.5T+3.48e3T2 1 - 51.5T + 3.48e3T^{2}
61 1+23.1iT3.72e3T2 1 + 23.1iT - 3.72e3T^{2}
67 1+113.T+4.48e3T2 1 + 113.T + 4.48e3T^{2}
71 18.00T+5.04e3T2 1 - 8.00T + 5.04e3T^{2}
73 1+32.5iT5.32e3T2 1 + 32.5iT - 5.32e3T^{2}
79 1+52.0iT6.24e3T2 1 + 52.0iT - 6.24e3T^{2}
83 143.3iT6.88e3T2 1 - 43.3iT - 6.88e3T^{2}
89 173.8T+7.92e3T2 1 - 73.8T + 7.92e3T^{2}
97 122.0T+9.40e3T2 1 - 22.0T + 9.40e3T^{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

−14.57000515499014124995599328951, −13.84579412680982573285077653821, −12.26338763133091248962446924422, −11.56688315435591868077543148778, −10.28596601990578832311633424353, −9.327769931827861834775259070508, −7.43969702475584404203980853587, −6.12679230052008586798972981634, −3.93831125514143753622104466982, −1.99565778490526600045696685395, 2.75584136954852601515315071505, 5.54713345154897930472551426192, 6.27908360837472805645640476979, 8.015837218665088524393195647623, 8.966561144628718457845937824634, 10.77666589082309009433420756659, 11.70402536067062103735555902640, 13.18446145816026083388920711875, 14.46523059179520259584166715042, 15.20109975396267457968008651279

Graph of the ZZ-function along the critical line