↑

Properties

Label 2-525-21.17-c1-0-3
Degree 22
Conductor 525525
Sign −0.709+0.704i-0.709 + 0.704i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.17 + 1.25i)2-s + (0.831 + 1.51i)3-s + (2.14 − 3.71i)4-s + (−3.71 − 2.25i)6-s + (−2.06 + 1.65i)7-s + 5.74i·8-s + (−1.61 + 2.52i)9-s + (1.48 + 0.859i)11-s + (7.42 + 0.172i)12-s + 0.360i·13-s + (2.39 − 6.18i)14-s + (−2.91 − 5.04i)16-s + (1.27 − 2.20i)17-s + (0.348 − 7.51i)18-s + (−4.93 + 2.84i)19-s + ⋯
L(s)  = 1  + (−1.53 + 0.886i)2-s + (0.479 + 0.877i)3-s + (1.07 − 1.85i)4-s + (−1.51 − 0.922i)6-s + (−0.778 + 0.627i)7-s + 2.03i·8-s + (−0.539 + 0.841i)9-s + (0.448 + 0.259i)11-s + (2.14 + 0.0496i)12-s + 0.100i·13-s + (0.639 − 1.65i)14-s + (−0.727 − 1.26i)16-s + (0.308 − 0.534i)17-s + (0.0821 − 1.77i)18-s + (−1.13 + 0.653i)19-s + ⋯

Functional equation

Λ(s)=(525s/2 ΓC(s) L(s)=((−0.709+0.704i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.709 + 0.704i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(525s/2 ΓC(s+1/2) L(s)=((−0.709+0.704i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.709 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 525525    =    3⋅52⋅73 \cdot 5^{2} \cdot 7
Sign: −0.709+0.704i-0.709 + 0.704i
Analytic conductor: 4.192144.19214
Root analytic conductor: 2.047472.04747
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ525(101,⋅)\chi_{525} (101, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 525, ( :1/2), −0.709+0.704i)(2,\ 525,\ (\ :1/2),\ -0.709 + 0.704i)

Particular Values

L(1)L(1) ≈\approx 0.149482−0.362731i0.149482 - 0.362731i
L(12)L(\frac12) ≈\approx 0.149482−0.362731i0.149482 - 0.362731i
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+(−0.831−1.51i)T 1 + (-0.831 - 1.51i)T
5 1 1
7 1+(2.06−1.65i)T 1 + (2.06 - 1.65i)T
good2 1+(2.17−1.25i)T+(1−1.73i)T2 1 + (2.17 - 1.25i)T + (1 - 1.73i)T^{2}
11 1+(−1.48−0.859i)T+(5.5+9.52i)T2 1 + (-1.48 - 0.859i)T + (5.5 + 9.52i)T^{2}
13 1−0.360iT−13T2 1 - 0.360iT - 13T^{2}
17 1+(−1.27+2.20i)T+(−8.5−14.7i)T2 1 + (-1.27 + 2.20i)T + (-8.5 - 14.7i)T^{2}
19 1+(4.93−2.84i)T+(9.5−16.4i)T2 1 + (4.93 - 2.84i)T + (9.5 - 16.4i)T^{2}
23 1+(2.17−1.25i)T+(11.5−19.9i)T2 1 + (2.17 - 1.25i)T + (11.5 - 19.9i)T^{2}
29 1−3.76iT−29T2 1 - 3.76iT - 29T^{2}
31 1+(2.41+1.39i)T+(15.5+26.8i)T2 1 + (2.41 + 1.39i)T + (15.5 + 26.8i)T^{2}
37 1+(−1.65−2.86i)T+(−18.5+32.0i)T2 1 + (-1.65 - 2.86i)T + (-18.5 + 32.0i)T^{2}
41 1+2.63T+41T2 1 + 2.63T + 41T^{2}
43 1+10.0T+43T2 1 + 10.0T + 43T^{2}
47 1+(2.91+5.04i)T+(−23.5+40.7i)T2 1 + (2.91 + 5.04i)T + (-23.5 + 40.7i)T^{2}
53 1+(1.25+0.727i)T+(26.5+45.8i)T2 1 + (1.25 + 0.727i)T + (26.5 + 45.8i)T^{2}
59 1+(−3.42+5.93i)T+(−29.5−51.0i)T2 1 + (-3.42 + 5.93i)T + (-29.5 - 51.0i)T^{2}
61 1+(−1.38+0.801i)T+(30.5−52.8i)T2 1 + (-1.38 + 0.801i)T + (30.5 - 52.8i)T^{2}
67 1+(−1.24+2.15i)T+(−33.5−58.0i)T2 1 + (-1.24 + 2.15i)T + (-33.5 - 58.0i)T^{2}
71 1−13.1iT−71T2 1 - 13.1iT - 71T^{2}
73 1+(−10.0−5.82i)T+(36.5+63.2i)T2 1 + (-10.0 - 5.82i)T + (36.5 + 63.2i)T^{2}
79 1+(6.93+12.0i)T+(−39.5+68.4i)T2 1 + (6.93 + 12.0i)T + (-39.5 + 68.4i)T^{2}
83 1+3.50T+83T2 1 + 3.50T + 83T^{2}
89 1+(−6.10−10.5i)T+(−44.5+77.0i)T2 1 + (-6.10 - 10.5i)T + (-44.5 + 77.0i)T^{2}
97 1−8.18iT−97T2 1 - 8.18iT - 97T^{2}
show more
show less
   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

−10.91684611671975863889333835649, −9.903692129758574426799835337445, −9.689173780869109927768161364215, −8.700353131873657000299216638948, −8.228891087483109902684892785998, −7.03636498674596702547383008283, −6.18136759013505849516424745429, −5.16848443371277691455970274891, −3.59184245103756387440802102177, −2.04886262377741627775584574434, 0.34442757927463772822488492569, 1.70527610884559296071080332416, 2.88758398862210930892233945866, 3.87484689739083384531414831908, 6.24808850351760258564436300809, 6.98108041514053396086455572724, 7.918348588567664368073021410354, 8.614166751717529524855046906725, 9.394602177539479459009695063279, 10.19211256144920385197620699211

Graph of the ZZ-function along the critical line