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Properties

Label 2-525-21.20-c1-0-39
Degree 22
Conductor 525525
Sign −0.0406+0.999i-0.0406 + 0.999i
Analytic cond. 4.192144.19214
Root an. cond. 2.047472.04747
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  − 0.792i·2-s + (1.18 − 1.26i)3-s + 1.37·4-s + (−1 − 0.939i)6-s + (2 − 1.73i)7-s − 2.67i·8-s + (−0.186 − 2.99i)9-s + 2.52i·11-s + (1.62 − 1.73i)12-s + 4.10i·13-s + (−1.37 − 1.58i)14-s + 0.627·16-s − 4.37·17-s + (−2.37 + 0.147i)18-s − 3.46i·19-s + ⋯
L(s)  = 1  − 0.560i·2-s + (0.684 − 0.728i)3-s + 0.686·4-s + (−0.408 − 0.383i)6-s + (0.755 − 0.654i)7-s − 0.944i·8-s + (−0.0620 − 0.998i)9-s + 0.761i·11-s + (0.469 − 0.499i)12-s + 1.13i·13-s + (−0.366 − 0.423i)14-s + 0.156·16-s − 1.06·17-s + (−0.559 + 0.0347i)18-s − 0.794i·19-s + ⋯

Functional equation

Λ(s)=(525s/2 ΓC(s) L(s)=((−0.0406+0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0406 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(525s/2 ΓC(s+1/2) L(s)=((−0.0406+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0406 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 525525    =    3⋅52⋅73 \cdot 5^{2} \cdot 7
Sign: −0.0406+0.999i-0.0406 + 0.999i
Analytic conductor: 4.192144.19214
Root analytic conductor: 2.047472.04747
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ525(251,⋅)\chi_{525} (251, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 525, ( :1/2), −0.0406+0.999i)(2,\ 525,\ (\ :1/2),\ -0.0406 + 0.999i)

Particular Values

L(1)L(1) ≈\approx 1.54774−1.61194i1.54774 - 1.61194i
L(12)L(\frac12) ≈\approx 1.54774−1.61194i1.54774 - 1.61194i
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)
bad3 1+(−1.18+1.26i)T 1 + (-1.18 + 1.26i)T
5 1 1
7 1+(−2+1.73i)T 1 + (-2 + 1.73i)T
good2 1+0.792iT−2T2 1 + 0.792iT - 2T^{2}
11 1−2.52iT−11T2 1 - 2.52iT - 11T^{2}
13 1−4.10iT−13T2 1 - 4.10iT - 13T^{2}
17 1+4.37T+17T2 1 + 4.37T + 17T^{2}
19 1+3.46iT−19T2 1 + 3.46iT - 19T^{2}
23 1−8.51iT−23T2 1 - 8.51iT - 23T^{2}
29 1+0.939iT−29T2 1 + 0.939iT - 29T^{2}
31 1−3.46iT−31T2 1 - 3.46iT - 31T^{2}
37 1−6.74T+37T2 1 - 6.74T + 37T^{2}
41 1+6T+41T2 1 + 6T + 41T^{2}
43 1+4.74T+43T2 1 + 4.74T + 43T^{2}
47 1−1.62T+47T2 1 - 1.62T + 47T^{2}
53 1+1.87iT−53T2 1 + 1.87iT - 53T^{2}
59 1+8.74T+59T2 1 + 8.74T + 59T^{2}
61 1−6.92iT−61T2 1 - 6.92iT - 61T^{2}
67 1+4.74T+67T2 1 + 4.74T + 67T^{2}
71 1+0.294iT−71T2 1 + 0.294iT - 71T^{2}
73 1+6.92iT−73T2 1 + 6.92iT - 73T^{2}
79 1+2.37T+79T2 1 + 2.37T + 79T^{2}
83 1−17.4T+83T2 1 - 17.4T + 83T^{2}
89 1−14.7T+89T2 1 - 14.7T + 89T^{2}
97 1−11.0iT−97T2 1 - 11.0iT - 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

−10.84410133330009901527984626773, −9.713086676263963082757410747299, −8.967002855571990760509489487900, −7.71404109989980366446907831645, −7.12089671327371578529796418527, −6.41934550863339423358096064731, −4.66036025455502599878801756939, −3.58518614421039986683783069661, −2.23246566136237335555898934150, −1.45756697331492100396381759157, 2.15558278360804170552666146541, 3.08884870290844587796562812934, 4.57903511795801412383980605138, 5.54141190815059891047079417856, 6.41433273282958980863982395942, 7.86285366798600350287657257982, 8.243978421817344486471213112731, 9.028939748744814991564033167835, 10.38610923828762090657650993622, 10.89048467268808551671247289692

Graph of the ZZ-function along the critical line