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Properties

Label 2-525-21.17-c1-0-27
Degree 22
Conductor 525525
Sign 0.921−0.389i0.921 - 0.389i
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  + (1.31 − 0.757i)2-s + (1.20 + 1.24i)3-s + (0.147 − 0.254i)4-s + (2.52 + 0.722i)6-s + (2.64 − 0.0753i)7-s + 2.58i·8-s + (−0.102 + 2.99i)9-s + (−1.86 − 1.07i)11-s + (0.494 − 0.123i)12-s − 3.48i·13-s + (3.41 − 2.10i)14-s + (2.25 + 3.89i)16-s + (−1.78 + 3.09i)17-s + (2.13 + 4.01i)18-s + (1.05 − 0.611i)19-s + ⋯
L(s)  = 1  + (0.927 − 0.535i)2-s + (0.694 + 0.719i)3-s + (0.0735 − 0.127i)4-s + (1.02 + 0.294i)6-s + (0.999 − 0.0284i)7-s + 0.913i·8-s + (−0.0341 + 0.999i)9-s + (−0.560 − 0.323i)11-s + (0.142 − 0.0356i)12-s − 0.965i·13-s + (0.911 − 0.561i)14-s + (0.562 + 0.974i)16-s + (−0.433 + 0.751i)17-s + (0.503 + 0.945i)18-s + (0.242 − 0.140i)19-s + ⋯

Functional equation

Λ(s)=(525s/2 ΓC(s) L(s)=((0.921−0.389i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.389i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(525s/2 ΓC(s+1/2) L(s)=((0.921−0.389i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 - 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 525525    =    3⋅52⋅73 \cdot 5^{2} \cdot 7
Sign: 0.921−0.389i0.921 - 0.389i
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), 0.921−0.389i)(2,\ 525,\ (\ :1/2),\ 0.921 - 0.389i)

Particular Values

L(1)L(1) ≈\approx 2.81896+0.571338i2.81896 + 0.571338i
L(12)L(\frac12) ≈\approx 2.81896+0.571338i2.81896 + 0.571338i
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+(−1.20−1.24i)T 1 + (-1.20 - 1.24i)T
5 1 1
7 1+(−2.64+0.0753i)T 1 + (-2.64 + 0.0753i)T
good2 1+(−1.31+0.757i)T+(1−1.73i)T2 1 + (-1.31 + 0.757i)T + (1 - 1.73i)T^{2}
11 1+(1.86+1.07i)T+(5.5+9.52i)T2 1 + (1.86 + 1.07i)T + (5.5 + 9.52i)T^{2}
13 1+3.48iT−13T2 1 + 3.48iT - 13T^{2}
17 1+(1.78−3.09i)T+(−8.5−14.7i)T2 1 + (1.78 - 3.09i)T + (-8.5 - 14.7i)T^{2}
19 1+(−1.05+0.611i)T+(9.5−16.4i)T2 1 + (-1.05 + 0.611i)T + (9.5 - 16.4i)T^{2}
23 1+(−1.31+0.757i)T+(11.5−19.9i)T2 1 + (-1.31 + 0.757i)T + (11.5 - 19.9i)T^{2}
29 1+5.95iT−29T2 1 + 5.95iT - 29T^{2}
31 1+(−2.75−1.58i)T+(15.5+26.8i)T2 1 + (-2.75 - 1.58i)T + (15.5 + 26.8i)T^{2}
37 1+(3.90+6.75i)T+(−18.5+32.0i)T2 1 + (3.90 + 6.75i)T + (-18.5 + 32.0i)T^{2}
41 1−11.8T+41T2 1 - 11.8T + 41T^{2}
43 1+2.99T+43T2 1 + 2.99T + 43T^{2}
47 1+(3.05+5.28i)T+(−23.5+40.7i)T2 1 + (3.05 + 5.28i)T + (-23.5 + 40.7i)T^{2}
53 1+(9.72+5.61i)T+(26.5+45.8i)T2 1 + (9.72 + 5.61i)T + (26.5 + 45.8i)T^{2}
59 1+(1.08−1.87i)T+(−29.5−51.0i)T2 1 + (1.08 - 1.87i)T + (-29.5 - 51.0i)T^{2}
61 1+(2.94−1.69i)T+(30.5−52.8i)T2 1 + (2.94 - 1.69i)T + (30.5 - 52.8i)T^{2}
67 1+(5.15−8.93i)T+(−33.5−58.0i)T2 1 + (5.15 - 8.93i)T + (-33.5 - 58.0i)T^{2}
71 1+10.3iT−71T2 1 + 10.3iT - 71T^{2}
73 1+(5.93+3.42i)T+(36.5+63.2i)T2 1 + (5.93 + 3.42i)T + (36.5 + 63.2i)T^{2}
79 1+(0.941+1.63i)T+(−39.5+68.4i)T2 1 + (0.941 + 1.63i)T + (-39.5 + 68.4i)T^{2}
83 1−9.10T+83T2 1 - 9.10T + 83T^{2}
89 1+(−0.889−1.54i)T+(−44.5+77.0i)T2 1 + (-0.889 - 1.54i)T + (-44.5 + 77.0i)T^{2}
97 1+1.32iT−97T2 1 + 1.32iT - 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

−10.86909125769726302037545564917, −10.42437279012310568484482156788, −9.062864532825734304508758770745, −8.201827313338823584535148191507, −7.72218954915785109089778384542, −5.77681478143583512440645399426, −4.95492420373905769824753244804, −4.16398267910670588571127194611, −3.13263245501839972410192611689, −2.14183592349895187898328334001, 1.47588335537145925488392212122, 2.91986094142310150706376437049, 4.31616307440206527206578042684, 5.05869022057840307216368470434, 6.26123420156294303254349798793, 7.12721414761851103780587661093, 7.81808284037691378440674014765, 8.931767351256546618637732903467, 9.694629705661101963755920510348, 11.01078394434062291840014956827

Graph of the ZZ-function along the critical line