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Properties

Label 2-525-105.62-c0-0-1
Degree 22
Conductor 525525
Sign 0.525−0.850i0.525 - 0.850i
Analytic cond. 0.2620090.262009
Root an. cond. 0.5118680.511868
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  + (0.707 + 0.707i)3-s + i·4-s + (0.707 − 0.707i)7-s + 1.00i·9-s + (−0.707 + 0.707i)12-s + (−1.41 − 1.41i)13-s − 16-s + 1.00·21-s + (−0.707 + 0.707i)27-s + (0.707 + 0.707i)28-s − 1.00·36-s − 2.00i·39-s + (−0.707 − 0.707i)48-s − 1.00i·49-s + (1.41 − 1.41i)52-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)3-s + i·4-s + (0.707 − 0.707i)7-s + 1.00i·9-s + (−0.707 + 0.707i)12-s + (−1.41 − 1.41i)13-s − 16-s + 1.00·21-s + (−0.707 + 0.707i)27-s + (0.707 + 0.707i)28-s − 1.00·36-s − 2.00i·39-s + (−0.707 − 0.707i)48-s − 1.00i·49-s + (1.41 − 1.41i)52-s + ⋯

Functional equation

Λ(s)=(525s/2 ΓC(s) L(s)=((0.525−0.850i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(525s/2 ΓC(s) L(s)=((0.525−0.850i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 525525    =    3⋅52⋅73 \cdot 5^{2} \cdot 7
Sign: 0.525−0.850i0.525 - 0.850i
Analytic conductor: 0.2620090.262009
Root analytic conductor: 0.5118680.511868
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ525(482,⋅)\chi_{525} (482, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 525, ( :0), 0.525−0.850i)(2,\ 525,\ (\ :0),\ 0.525 - 0.850i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0990144801.099014480
L(12)L(\frac12) ≈\approx 1.0990144801.099014480
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+(−0.707−0.707i)T 1 + (-0.707 - 0.707i)T
5 1 1
7 1+(−0.707+0.707i)T 1 + (-0.707 + 0.707i)T
good2 1−iT2 1 - iT^{2}
11 1−T2 1 - T^{2}
13 1+(1.41+1.41i)T+iT2 1 + (1.41 + 1.41i)T + iT^{2}
17 1+iT2 1 + iT^{2}
19 1+T2 1 + T^{2}
23 1+iT2 1 + iT^{2}
29 1+T2 1 + T^{2}
31 1−T2 1 - T^{2}
37 1+iT2 1 + iT^{2}
41 1+T2 1 + T^{2}
43 1−iT2 1 - iT^{2}
47 1+iT2 1 + iT^{2}
53 1+iT2 1 + iT^{2}
59 1−T2 1 - T^{2}
61 1−T2 1 - T^{2}
67 1+iT2 1 + iT^{2}
71 1−T2 1 - T^{2}
73 1+(−1.41−1.41i)T+iT2 1 + (-1.41 - 1.41i)T + iT^{2}
79 1+2iT−T2 1 + 2iT - T^{2}
83 1−iT2 1 - iT^{2}
89 1−T2 1 - T^{2}
97 1+(1.41−1.41i)T−iT2 1 + (1.41 - 1.41i)T - iT^{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

−11.06108380627880486390462260870, −10.30225557912887942282927169291, −9.464701425020300512291751869324, −8.346892619081897966562496969315, −7.81686311572672741583771313411, −7.12229114070007169020647589642, −5.25884991901432943458564428020, −4.44842047778893738728465787677, −3.41988749853379855521739311917, −2.42039816419682535206851573929, 1.69822973109905766515715123762, 2.51579122388132870600032895965, 4.36040654690935255497014599943, 5.35157498113411266144568729314, 6.48035186241955739564708130023, 7.24952941496681362549494908476, 8.326894639642039016689801821975, 9.257667905558522955721180684344, 9.726748370637548568561613724359, 11.04577503457517942785670812698

Graph of the ZZ-function along the critical line