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Properties

Label 2-2664-37.10-c1-0-5
Degree 22
Conductor 26642664
Sign −0.648−0.761i-0.648 - 0.761i
Analytic cond. 21.272121.2721
Root an. cond. 4.612174.61217
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  + (−1.87 − 3.24i)5-s + (2.03 + 3.52i)7-s − 1.66·11-s + (0.106 + 0.184i)13-s + (−0.5 + 0.866i)17-s + (−0.271 − 0.470i)19-s + 4.61·23-s + (−4.51 + 7.82i)25-s − 9.65·29-s − 1.66·31-s + (7.62 − 13.2i)35-s + (−3.22 − 5.16i)37-s + (−0.729 − 1.26i)41-s − 2.74·43-s − 3.89·47-s + ⋯
L(s)  = 1  + (−0.837 − 1.45i)5-s + (0.769 + 1.33i)7-s − 0.503·11-s + (0.0295 + 0.0511i)13-s + (−0.121 + 0.210i)17-s + (−0.0623 − 0.107i)19-s + 0.962·23-s + (−0.903 + 1.56i)25-s − 1.79·29-s − 0.299·31-s + (1.28 − 2.23i)35-s + (−0.529 − 0.848i)37-s + (−0.113 − 0.197i)41-s − 0.418·43-s − 0.568·47-s + ⋯

Functional equation

Λ(s)=(2664s/2 ΓC(s) L(s)=((−0.648−0.761i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.648 - 0.761i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2664s/2 ΓC(s+1/2) L(s)=((−0.648−0.761i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.648 - 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 26642664    =    23⋅32⋅372^{3} \cdot 3^{2} \cdot 37
Sign: −0.648−0.761i-0.648 - 0.761i
Analytic conductor: 21.272121.2721
Root analytic conductor: 4.612174.61217
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2664(1009,⋅)\chi_{2664} (1009, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2664, ( :1/2), −0.648−0.761i)(2,\ 2664,\ (\ :1/2),\ -0.648 - 0.761i)

Particular Values

L(1)L(1) ≈\approx 0.41194167100.4119416710
L(12)L(\frac12) ≈\approx 0.41194167100.4119416710
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
37 1+(3.22+5.16i)T 1 + (3.22 + 5.16i)T
good5 1+(1.87+3.24i)T+(−2.5+4.33i)T2 1 + (1.87 + 3.24i)T + (-2.5 + 4.33i)T^{2}
7 1+(−2.03−3.52i)T+(−3.5+6.06i)T2 1 + (-2.03 - 3.52i)T + (-3.5 + 6.06i)T^{2}
11 1+1.66T+11T2 1 + 1.66T + 11T^{2}
13 1+(−0.106−0.184i)T+(−6.5+11.2i)T2 1 + (-0.106 - 0.184i)T + (-6.5 + 11.2i)T^{2}
17 1+(0.5−0.866i)T+(−8.5−14.7i)T2 1 + (0.5 - 0.866i)T + (-8.5 - 14.7i)T^{2}
19 1+(0.271+0.470i)T+(−9.5+16.4i)T2 1 + (0.271 + 0.470i)T + (-9.5 + 16.4i)T^{2}
23 1−4.61T+23T2 1 - 4.61T + 23T^{2}
29 1+9.65T+29T2 1 + 9.65T + 29T^{2}
31 1+1.66T+31T2 1 + 1.66T + 31T^{2}
41 1+(0.729+1.26i)T+(−20.5+35.5i)T2 1 + (0.729 + 1.26i)T + (-20.5 + 35.5i)T^{2}
43 1+2.74T+43T2 1 + 2.74T + 43T^{2}
47 1+3.89T+47T2 1 + 3.89T + 47T^{2}
53 1+(3.86−6.69i)T+(−26.5−45.8i)T2 1 + (3.86 - 6.69i)T + (-26.5 - 45.8i)T^{2}
59 1+(−0.563+0.975i)T+(−29.5−51.0i)T2 1 + (-0.563 + 0.975i)T + (-29.5 - 51.0i)T^{2}
61 1+(−2.85−4.95i)T+(−30.5+52.8i)T2 1 + (-2.85 - 4.95i)T + (-30.5 + 52.8i)T^{2}
67 1+(3.01+5.22i)T+(−33.5+58.0i)T2 1 + (3.01 + 5.22i)T + (-33.5 + 58.0i)T^{2}
71 1+(−3.49−6.05i)T+(−35.5+61.4i)T2 1 + (-3.49 - 6.05i)T + (-35.5 + 61.4i)T^{2}
73 1−2.12T+73T2 1 - 2.12T + 73T^{2}
79 1+(−4.77−8.27i)T+(−39.5+68.4i)T2 1 + (-4.77 - 8.27i)T + (-39.5 + 68.4i)T^{2}
83 1+(6.22−10.7i)T+(−41.5−71.8i)T2 1 + (6.22 - 10.7i)T + (-41.5 - 71.8i)T^{2}
89 1+(2.72−4.72i)T+(−44.5−77.0i)T2 1 + (2.72 - 4.72i)T + (-44.5 - 77.0i)T^{2}
97 1+9.99T+97T2 1 + 9.99T + 97T^{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

−9.068705279174907420784985276124, −8.345281669393610167276172889117, −7.922278261798383152107200968022, −7.00591261318680721115560553392, −5.59581235284750078582272780976, −5.32625541770788964693322956346, −4.54169240213270974247844912926, −3.63475475489356609995755885780, −2.35089800590800378149833593014, −1.36123029326194816673563989172, 0.13537316502969334761080629619, 1.71848423897861483772371747724, 3.02385409837256913777035877277, 3.65257745328208434445906413192, 4.47941103392809050505712006633, 5.38119438952972808538621025543, 6.65820431231124756928545820751, 7.08934960012281132223244585264, 7.73618134472575958645295804138, 8.223499772622360641345494103932

Graph of the ZZ-function along the critical line