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Properties

Label 2-15e3-75.29-c0-0-2
Degree 22
Conductor 33753375
Sign 0.990+0.137i0.990 + 0.137i
Analytic cond. 1.684341.68434
Root an. cond. 1.297821.29782
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  + (0.190 − 0.587i)2-s + (0.5 + 0.363i)4-s + 0.618i·7-s + (0.809 − 0.587i)8-s + (0.951 + 0.309i)11-s + (0.363 + 0.118i)14-s + (0.809 − 0.587i)17-s + (−0.809 + 0.587i)19-s + (0.363 − 0.5i)22-s + (0.309 − 0.951i)23-s + (−0.224 + 0.309i)28-s + (−0.951 + 1.30i)29-s + 32-s + (−0.190 − 0.587i)34-s + (−1.53 + 0.5i)37-s + (0.190 + 0.587i)38-s + ⋯
L(s)  = 1  + (0.190 − 0.587i)2-s + (0.5 + 0.363i)4-s + 0.618i·7-s + (0.809 − 0.587i)8-s + (0.951 + 0.309i)11-s + (0.363 + 0.118i)14-s + (0.809 − 0.587i)17-s + (−0.809 + 0.587i)19-s + (0.363 − 0.5i)22-s + (0.309 − 0.951i)23-s + (−0.224 + 0.309i)28-s + (−0.951 + 1.30i)29-s + 32-s + (−0.190 − 0.587i)34-s + (−1.53 + 0.5i)37-s + (0.190 + 0.587i)38-s + ⋯

Functional equation

Λ(s)=(3375s/2 ΓC(s) L(s)=((0.990+0.137i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3375s/2 ΓC(s) L(s)=((0.990+0.137i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33753375    =    33⋅533^{3} \cdot 5^{3}
Sign: 0.990+0.137i0.990 + 0.137i
Analytic conductor: 1.684341.68434
Root analytic conductor: 1.297821.29782
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3375(2024,⋅)\chi_{3375} (2024, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3375, ( :0), 0.990+0.137i)(2,\ 3375,\ (\ :0),\ 0.990 + 0.137i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.8147532591.814753259
L(12)L(\frac12) ≈\approx 1.8147532591.814753259
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
5 1 1
good2 1+(−0.190+0.587i)T+(−0.809−0.587i)T2 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2}
7 1−0.618iT−T2 1 - 0.618iT - T^{2}
11 1+(−0.951−0.309i)T+(0.809+0.587i)T2 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2}
13 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
17 1+(−0.809+0.587i)T+(0.309−0.951i)T2 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2}
19 1+(0.809−0.587i)T+(0.309−0.951i)T2 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2}
23 1+(−0.309+0.951i)T+(−0.809−0.587i)T2 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2}
29 1+(0.951−1.30i)T+(−0.309−0.951i)T2 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2}
31 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
37 1+(1.53−0.5i)T+(0.809−0.587i)T2 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2}
41 1+(−0.951+0.309i)T+(0.809−0.587i)T2 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2}
43 1−T2 1 - T^{2}
47 1+(1.30+0.951i)T+(0.309+0.951i)T2 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2}
53 1+(−0.5−0.363i)T+(0.309+0.951i)T2 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2}
59 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
61 1+(0.309−0.951i)T+(−0.809−0.587i)T2 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2}
67 1+(0.363+0.5i)T+(−0.309+0.951i)T2 1 + (0.363 + 0.5i)T + (-0.309 + 0.951i)T^{2}
71 1+(−0.951+1.30i)T+(−0.309−0.951i)T2 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2}
73 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
79 1+(−1.30−0.951i)T+(0.309+0.951i)T2 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2}
83 1+(0.5−0.363i)T+(0.309−0.951i)T2 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2}
89 1+(0.951+0.309i)T+(0.809+0.587i)T2 1 + (0.951 + 0.309i)T + (0.809 + 0.587i)T^{2}
97 1+(−0.587+0.809i)T+(−0.309−0.951i)T2 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)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

−8.815986022271907191301223247750, −8.097557626405148496907089449777, −7.13815426204121128206874913953, −6.69968513588744557734505972401, −5.75104921929179846135189755015, −4.84288962609552941080537534787, −3.89023645500129368226229263369, −3.23193908112851736057354896403, −2.25252821613182854163250819550, −1.43160450307850129520232372989, 1.18712356863815812952797011067, 2.15155011242515581392475843215, 3.47778989169615192267272742265, 4.19562795238856637335805730037, 5.17573704514736112174493575727, 5.95486752473156564757581521308, 6.51463640767243350767248234583, 7.28372089157090954529625760987, 7.82376882045222861586065657412, 8.694963801476350562316918180464

Graph of the ZZ-function along the critical line