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Properties

Label 2-873-97.46-c0-0-0
Degree 22
Conductor 873873
Sign 0.637+0.770i0.637 + 0.770i
Analytic cond. 0.4356830.435683
Root an. cond. 0.6600630.660063
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  + (0.382 − 0.923i)4-s + (1.68 − 0.902i)7-s + (−1.47 + 0.448i)13-s + (−0.707 − 0.707i)16-s + (−0.831 + 1.55i)19-s + (−0.555 − 0.831i)25-s + (−0.187 − 1.90i)28-s + (0.636 + 0.425i)31-s + (1.53 + 1.26i)37-s + (0.149 − 0.360i)43-s + (1.47 − 2.21i)49-s + (−0.151 + 1.53i)52-s − 1.96·61-s + (−0.923 + 0.382i)64-s + (0.187 − 0.0569i)67-s + ⋯
L(s)  = 1  + (0.382 − 0.923i)4-s + (1.68 − 0.902i)7-s + (−1.47 + 0.448i)13-s + (−0.707 − 0.707i)16-s + (−0.831 + 1.55i)19-s + (−0.555 − 0.831i)25-s + (−0.187 − 1.90i)28-s + (0.636 + 0.425i)31-s + (1.53 + 1.26i)37-s + (0.149 − 0.360i)43-s + (1.47 − 2.21i)49-s + (−0.151 + 1.53i)52-s − 1.96·61-s + (−0.923 + 0.382i)64-s + (0.187 − 0.0569i)67-s + ⋯

Functional equation

Λ(s)=(873s/2 ΓC(s) L(s)=((0.637+0.770i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 873 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(873s/2 ΓC(s) L(s)=((0.637+0.770i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 873 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 873873    =    32⋅973^{2} \cdot 97
Sign: 0.637+0.770i0.637 + 0.770i
Analytic conductor: 0.4356830.435683
Root analytic conductor: 0.6600630.660063
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ873(46,⋅)\chi_{873} (46, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 873, ( :0), 0.637+0.770i)(2,\ 873,\ (\ :0),\ 0.637 + 0.770i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.1746016511.174601651
L(12)L(\frac12) ≈\approx 1.1746016511.174601651
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)
bad3 1 1
97 1+(−0.555−0.831i)T 1 + (-0.555 - 0.831i)T
good2 1+(−0.382+0.923i)T2 1 + (-0.382 + 0.923i)T^{2}
5 1+(0.555+0.831i)T2 1 + (0.555 + 0.831i)T^{2}
7 1+(−1.68+0.902i)T+(0.555−0.831i)T2 1 + (-1.68 + 0.902i)T + (0.555 - 0.831i)T^{2}
11 1+(−0.923+0.382i)T2 1 + (-0.923 + 0.382i)T^{2}
13 1+(1.47−0.448i)T+(0.831−0.555i)T2 1 + (1.47 - 0.448i)T + (0.831 - 0.555i)T^{2}
17 1+(0.831−0.555i)T2 1 + (0.831 - 0.555i)T^{2}
19 1+(0.831−1.55i)T+(−0.555−0.831i)T2 1 + (0.831 - 1.55i)T + (-0.555 - 0.831i)T^{2}
23 1+(0.980+0.195i)T2 1 + (0.980 + 0.195i)T^{2}
29 1+(0.980+0.195i)T2 1 + (0.980 + 0.195i)T^{2}
31 1+(−0.636−0.425i)T+(0.382+0.923i)T2 1 + (-0.636 - 0.425i)T + (0.382 + 0.923i)T^{2}
37 1+(−1.53−1.26i)T+(0.195+0.980i)T2 1 + (-1.53 - 1.26i)T + (0.195 + 0.980i)T^{2}
41 1+(−0.195+0.980i)T2 1 + (-0.195 + 0.980i)T^{2}
43 1+(−0.149+0.360i)T+(−0.707−0.707i)T2 1 + (-0.149 + 0.360i)T + (-0.707 - 0.707i)T^{2}
47 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)T^{2}
53 1+(−0.923−0.382i)T2 1 + (-0.923 - 0.382i)T^{2}
59 1+(−0.980+0.195i)T2 1 + (-0.980 + 0.195i)T^{2}
61 1+1.96T+T2 1 + 1.96T + T^{2}
67 1+(−0.187+0.0569i)T+(0.831−0.555i)T2 1 + (-0.187 + 0.0569i)T + (0.831 - 0.555i)T^{2}
71 1+(−0.195−0.980i)T2 1 + (-0.195 - 0.980i)T^{2}
73 1+(−0.541−1.30i)T+(−0.707+0.707i)T2 1 + (-0.541 - 1.30i)T + (-0.707 + 0.707i)T^{2}
79 1+(0.617−0.923i)T+(−0.382−0.923i)T2 1 + (0.617 - 0.923i)T + (-0.382 - 0.923i)T^{2}
83 1+(−0.555−0.831i)T2 1 + (-0.555 - 0.831i)T^{2}
89 1+(−0.923+0.382i)T2 1 + (-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

−10.23089681101037697223022650146, −9.742061762025986235146106771191, −8.346091591860330697396174916271, −7.72502997015733101884135307442, −6.85022255975039753392156907492, −5.84653616396044699292887825970, −4.78144690721236767716156081201, −4.28859248049170599807089921917, −2.34220287757922849915631445119, −1.40097538325059712507504642898, 2.10709150590364125851547344004, 2.74930528932712349599109539927, 4.38822877483185302351965633723, 4.99366016905611748692114074335, 6.13793847489178212257811125486, 7.48463624856851664682602167948, 7.74084385723075848434965038593, 8.730408312712736030206437243384, 9.386653708552690928370765302291, 10.79113881037889470115347097677

Graph of the ZZ-function along the critical line