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Properties

Label 2-405-45.34-c1-0-12
Degree 22
Conductor 405405
Sign −0.476+0.879i-0.476 + 0.879i
Analytic cond. 3.233943.23394
Root an. cond. 1.798311.79831
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  + (−1.22 − 0.707i)2-s + (−0.448 − 2.19i)5-s + (2.59 + 1.5i)7-s + 2.82i·8-s + (−1 + 2.99i)10-s + (2.12 − 3.67i)11-s + (2.59 − 1.5i)13-s + (−2.12 − 3.67i)14-s + (2.00 − 3.46i)16-s − 2.82i·17-s − 19-s + (−5.19 + 3i)22-s + (−6.12 + 3.53i)23-s + (−4.59 + 1.96i)25-s − 4.24·26-s + ⋯
L(s)  = 1  + (−0.866 − 0.499i)2-s + (−0.200 − 0.979i)5-s + (0.981 + 0.566i)7-s + 0.999i·8-s + (−0.316 + 0.948i)10-s + (0.639 − 1.10i)11-s + (0.720 − 0.416i)13-s + (−0.566 − 0.981i)14-s + (0.500 − 0.866i)16-s − 0.685i·17-s − 0.229·19-s + (−1.10 + 0.639i)22-s + (−1.27 + 0.737i)23-s + (−0.919 + 0.392i)25-s − 0.832·26-s + ⋯

Functional equation

Λ(s)=(405s/2 ΓC(s) L(s)=((−0.476+0.879i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.476 + 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(405s/2 ΓC(s+1/2) L(s)=((−0.476+0.879i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.476 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 405405    =    34⋅53^{4} \cdot 5
Sign: −0.476+0.879i-0.476 + 0.879i
Analytic conductor: 3.233943.23394
Root analytic conductor: 1.798311.79831
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ405(379,⋅)\chi_{405} (379, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 405, ( :1/2), −0.476+0.879i)(2,\ 405,\ (\ :1/2),\ -0.476 + 0.879i)

Particular Values

L(1)L(1) ≈\approx 0.430296−0.722329i0.430296 - 0.722329i
L(12)L(\frac12) ≈\approx 0.430296−0.722329i0.430296 - 0.722329i
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
5 1+(0.448+2.19i)T 1 + (0.448 + 2.19i)T
good2 1+(1.22+0.707i)T+(1+1.73i)T2 1 + (1.22 + 0.707i)T + (1 + 1.73i)T^{2}
7 1+(−2.59−1.5i)T+(3.5+6.06i)T2 1 + (-2.59 - 1.5i)T + (3.5 + 6.06i)T^{2}
11 1+(−2.12+3.67i)T+(−5.5−9.52i)T2 1 + (-2.12 + 3.67i)T + (-5.5 - 9.52i)T^{2}
13 1+(−2.59+1.5i)T+(6.5−11.2i)T2 1 + (-2.59 + 1.5i)T + (6.5 - 11.2i)T^{2}
17 1+2.82iT−17T2 1 + 2.82iT - 17T^{2}
19 1+T+19T2 1 + T + 19T^{2}
23 1+(6.12−3.53i)T+(11.5−19.9i)T2 1 + (6.12 - 3.53i)T + (11.5 - 19.9i)T^{2}
29 1+(−2.12+3.67i)T+(−14.5−25.1i)T2 1 + (-2.12 + 3.67i)T + (-14.5 - 25.1i)T^{2}
31 1+(1+1.73i)T+(−15.5+26.8i)T2 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2}
37 1+9iT−37T2 1 + 9iT - 37T^{2}
41 1+(2.12+3.67i)T+(−20.5+35.5i)T2 1 + (2.12 + 3.67i)T + (-20.5 + 35.5i)T^{2}
43 1+(5.19+3i)T+(21.5+37.2i)T2 1 + (5.19 + 3i)T + (21.5 + 37.2i)T^{2}
47 1+(−2.44−1.41i)T+(23.5+40.7i)T2 1 + (-2.44 - 1.41i)T + (23.5 + 40.7i)T^{2}
53 1−9.89iT−53T2 1 - 9.89iT - 53T^{2}
59 1+(−4.24−7.34i)T+(−29.5+51.0i)T2 1 + (-4.24 - 7.34i)T + (-29.5 + 51.0i)T^{2}
61 1+(−6.5+11.2i)T+(−30.5−52.8i)T2 1 + (-6.5 + 11.2i)T + (-30.5 - 52.8i)T^{2}
67 1+(−2.59+1.5i)T+(33.5−58.0i)T2 1 + (-2.59 + 1.5i)T + (33.5 - 58.0i)T^{2}
71 1−12.7T+71T2 1 - 12.7T + 71T^{2}
73 1−9iT−73T2 1 - 9iT - 73T^{2}
79 1+(2.5−4.33i)T+(−39.5−68.4i)T2 1 + (2.5 - 4.33i)T + (-39.5 - 68.4i)T^{2}
83 1+(1.22+0.707i)T+(41.5+71.8i)T2 1 + (1.22 + 0.707i)T + (41.5 + 71.8i)T^{2}
89 1+89T2 1 + 89T^{2}
97 1+(−2.59−1.5i)T+(48.5+84.0i)T2 1 + (-2.59 - 1.5i)T + (48.5 + 84.0i)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

−11.13909164091295481024589458278, −9.914641389458017228359356276668, −8.984296637690214887112137100129, −8.482691400341184848801660615222, −7.80453292734435524920609826251, −5.88914286965259600759547521591, −5.25277899724708434073001718399, −3.90402059407269633201049804318, −2.04446764909690950970508946416, −0.801129662420497264970198870229, 1.68823999176220550534605275916, 3.67148859113993749577723387597, 4.51390673864806631627567795241, 6.44861066936021542415933671987, 6.92426673840184250560182982203, 7.991625300352476129432568636045, 8.500906199013635890535999946627, 9.832849069882315832683557039177, 10.38833015051355673899477000273, 11.38775438797652661230251385468

Graph of the ZZ-function along the critical line