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Properties

Label 2-3332-476.199-c0-0-1
Degree 22
Conductor 33323332
Sign 0.893−0.448i0.893 - 0.448i
Analytic cond. 1.662881.66288
Root an. cond. 1.289521.28952
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.608 + 0.793i)2-s + (−0.258 − 0.965i)4-s + (1.05 − 0.357i)5-s + (0.923 + 0.382i)8-s + (0.991 − 0.130i)9-s + (−0.357 + 1.05i)10-s + (1 − i)13-s + (−0.866 + 0.499i)16-s + (−0.130 + 0.991i)17-s + (−0.499 + 0.866i)18-s + (−0.617 − 0.923i)20-s + (0.186 − 0.142i)25-s + (0.184 + 1.40i)26-s + (−0.216 + 1.08i)29-s + (0.130 − 0.991i)32-s + ⋯
L(s)  = 1  + (−0.608 + 0.793i)2-s + (−0.258 − 0.965i)4-s + (1.05 − 0.357i)5-s + (0.923 + 0.382i)8-s + (0.991 − 0.130i)9-s + (−0.357 + 1.05i)10-s + (1 − i)13-s + (−0.866 + 0.499i)16-s + (−0.130 + 0.991i)17-s + (−0.499 + 0.866i)18-s + (−0.617 − 0.923i)20-s + (0.186 − 0.142i)25-s + (0.184 + 1.40i)26-s + (−0.216 + 1.08i)29-s + (0.130 − 0.991i)32-s + ⋯

Functional equation

Λ(s)=(3332s/2 ΓC(s) L(s)=((0.893−0.448i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3332s/2 ΓC(s) L(s)=((0.893−0.448i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33323332    =    22⋅72⋅172^{2} \cdot 7^{2} \cdot 17
Sign: 0.893−0.448i0.893 - 0.448i
Analytic conductor: 1.662881.66288
Root analytic conductor: 1.289521.28952
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3332(2579,⋅)\chi_{3332} (2579, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3332, ( :0), 0.893−0.448i)(2,\ 3332,\ (\ :0),\ 0.893 - 0.448i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.2604903411.260490341
L(12)L(\frac12) ≈\approx 1.2604903411.260490341
L(1)L(1) 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+(0.608−0.793i)T 1 + (0.608 - 0.793i)T
7 1 1
17 1+(0.130−0.991i)T 1 + (0.130 - 0.991i)T
good3 1+(−0.991+0.130i)T2 1 + (-0.991 + 0.130i)T^{2}
5 1+(−1.05+0.357i)T+(0.793−0.608i)T2 1 + (-1.05 + 0.357i)T + (0.793 - 0.608i)T^{2}
11 1+(0.608−0.793i)T2 1 + (0.608 - 0.793i)T^{2}
13 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{2}
19 1+(−0.258−0.965i)T2 1 + (-0.258 - 0.965i)T^{2}
23 1+(0.991+0.130i)T2 1 + (0.991 + 0.130i)T^{2}
29 1+(0.216−1.08i)T+(−0.923−0.382i)T2 1 + (0.216 - 1.08i)T + (-0.923 - 0.382i)T^{2}
31 1+(0.991−0.130i)T2 1 + (0.991 - 0.130i)T^{2}
37 1+(−1.49−0.735i)T+(0.608+0.793i)T2 1 + (-1.49 - 0.735i)T + (0.608 + 0.793i)T^{2}
41 1+(0.0761+0.382i)T+(−0.923+0.382i)T2 1 + (0.0761 + 0.382i)T + (-0.923 + 0.382i)T^{2}
43 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)T^{2}
47 1+(−0.866−0.5i)T2 1 + (-0.866 - 0.5i)T^{2}
53 1+(1.83+0.241i)T+(0.965+0.258i)T2 1 + (1.83 + 0.241i)T + (0.965 + 0.258i)T^{2}
59 1+(0.258−0.965i)T2 1 + (0.258 - 0.965i)T^{2}
61 1+(1.29+1.47i)T+(−0.130+0.991i)T2 1 + (1.29 + 1.47i)T + (-0.130 + 0.991i)T^{2}
67 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
71 1+(−0.382+0.923i)T2 1 + (-0.382 + 0.923i)T^{2}
73 1+(1.25+1.09i)T+(0.130+0.991i)T2 1 + (1.25 + 1.09i)T + (0.130 + 0.991i)T^{2}
79 1+(−0.991−0.130i)T2 1 + (-0.991 - 0.130i)T^{2}
83 1+(0.707−0.707i)T2 1 + (0.707 - 0.707i)T^{2}
89 1+(−0.866−0.5i)T2 1 + (-0.866 - 0.5i)T^{2}
97 1+(−1.63−0.324i)T+(0.923+0.382i)T2 1 + (-1.63 - 0.324i)T + (0.923 + 0.382i)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

−8.925911487940695155658090908969, −8.065966568746956959050157990284, −7.52391157952636980027985164993, −6.33038074695391050446788874403, −6.19442592356842810173246289385, −5.25267308233358990546222721439, −4.51876462746256053539642016659, −3.38809986507536881825822977876, −1.83811639308141362873532887781, −1.20835503897156078869665729075, 1.26242599526066894770155284304, 2.06778309368915818642433088642, 2.89000326733841961606683478413, 4.06412250101464644010068383183, 4.60525947405653588351217057143, 5.89684590354402991110130771020, 6.59635666399401870136614512860, 7.36920567138313603483294043605, 8.076528755421906125418161609008, 9.220277042375867753857532785913

Graph of the ZZ-function along the critical line