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Properties

Label 2-60e2-100.31-c0-0-1
Degree 22
Conductor 36003600
Sign 0.637+0.770i0.637 + 0.770i
Analytic cond. 1.796631.79663
Root an. cond. 1.340381.34038
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  + (0.587 − 0.809i)5-s + (0.5 − 1.53i)13-s + (1.53 + 1.11i)17-s + (−0.309 − 0.951i)25-s + (−1.53 + 1.11i)29-s + (0.190 − 0.587i)37-s + (0.363 − 1.11i)41-s + 49-s + (−0.951 + 0.690i)53-s + (−0.5 − 1.53i)61-s + (−0.951 − 1.30i)65-s + (0.5 + 1.53i)73-s + (1.80 − 0.587i)85-s + (−0.587 − 1.80i)89-s + (0.5 − 0.363i)97-s + ⋯
L(s)  = 1  + (0.587 − 0.809i)5-s + (0.5 − 1.53i)13-s + (1.53 + 1.11i)17-s + (−0.309 − 0.951i)25-s + (−1.53 + 1.11i)29-s + (0.190 − 0.587i)37-s + (0.363 − 1.11i)41-s + 49-s + (−0.951 + 0.690i)53-s + (−0.5 − 1.53i)61-s + (−0.951 − 1.30i)65-s + (0.5 + 1.53i)73-s + (1.80 − 0.587i)85-s + (−0.587 − 1.80i)89-s + (0.5 − 0.363i)97-s + ⋯

Functional equation

Λ(s)=(3600s/2 ΓC(s) L(s)=((0.637+0.770i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3600s/2 ΓC(s) L(s)=((0.637+0.770i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 36003600    =    24⋅32⋅522^{4} \cdot 3^{2} \cdot 5^{2}
Sign: 0.637+0.770i0.637 + 0.770i
Analytic conductor: 1.796631.79663
Root analytic conductor: 1.340381.34038
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3600(2431,⋅)\chi_{3600} (2431, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3600, ( :0), 0.637+0.770i)(2,\ 3600,\ (\ :0),\ 0.637 + 0.770i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.4915388941.491538894
L(12)L(\frac12) ≈\approx 1.4915388941.491538894
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 1
3 1 1
5 1+(−0.587+0.809i)T 1 + (-0.587 + 0.809i)T
good7 1−T2 1 - T^{2}
11 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
13 1+(−0.5+1.53i)T+(−0.809−0.587i)T2 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2}
17 1+(−1.53−1.11i)T+(0.309+0.951i)T2 1 + (-1.53 - 1.11i)T + (0.309 + 0.951i)T^{2}
19 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
23 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
29 1+(1.53−1.11i)T+(0.309−0.951i)T2 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2}
31 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
37 1+(−0.190+0.587i)T+(−0.809−0.587i)T2 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2}
41 1+(−0.363+1.11i)T+(−0.809−0.587i)T2 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2}
43 1−T2 1 - T^{2}
47 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
53 1+(0.951−0.690i)T+(0.309−0.951i)T2 1 + (0.951 - 0.690i)T + (0.309 - 0.951i)T^{2}
59 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
61 1+(0.5+1.53i)T+(−0.809+0.587i)T2 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2}
67 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
71 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
73 1+(−0.5−1.53i)T+(−0.809+0.587i)T2 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2}
79 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
83 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
89 1+(0.587+1.80i)T+(−0.809+0.587i)T2 1 + (0.587 + 1.80i)T + (-0.809 + 0.587i)T^{2}
97 1+(−0.5+0.363i)T+(0.309−0.951i)T2 1 + (-0.5 + 0.363i)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

−8.634645538723734877188175634893, −7.928415303651406930754436133120, −7.36004432837544540425262262356, −6.02599320117936753597049595904, −5.69878666478171140842910200569, −5.06112316946891337130104262141, −3.87668915477515962908998267055, −3.21793009496791806389229853546, −1.92708906652448441659525863001, −0.977557961465797295495772041483, 1.41737282729308541845354976111, 2.39859839011044621208651091764, 3.30106422423217029575374342755, 4.13633216876818177349587965202, 5.16233490095659198185152732386, 5.96568153634088584923879080575, 6.55436437929771469128905478995, 7.37102695531173457431326062567, 7.893111300711987573229854036254, 9.108364717096661180817865307297

Graph of the ZZ-function along the critical line