↑

Properties

Label 2-325-5.4-c5-0-5
Degree 22
Conductor 325325
Sign −0.447+0.894i-0.447 + 0.894i
Analytic cond. 52.124752.1247
Root an. cond. 7.219747.21974
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.96i·2-s + 29.7i·3-s + 28.1·4-s + 58.4·6-s + 176. i·7-s − 118. i·8-s − 643.·9-s − 62.7·11-s + 837. i·12-s − 169i·13-s + 347.·14-s + 668.·16-s − 1.82e3i·17-s + 1.26e3i·18-s − 2.50e3·19-s + ⋯
L(s)  = 1  − 0.347i·2-s + 1.90i·3-s + 0.879·4-s + 0.663·6-s + 1.36i·7-s − 0.652i·8-s − 2.64·9-s − 0.156·11-s + 1.67i·12-s − 0.277i·13-s + 0.474·14-s + 0.652·16-s − 1.53i·17-s + 0.919i·18-s − 1.59·19-s + ⋯

Functional equation

Λ(s)=(325s/2 ΓC(s) L(s)=((−0.447+0.894i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(325s/2 ΓC(s+5/2) L(s)=((−0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 325325    =    52⋅135^{2} \cdot 13
Sign: −0.447+0.894i-0.447 + 0.894i
Analytic conductor: 52.124752.1247
Root analytic conductor: 7.219747.21974
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ325(274,⋅)\chi_{325} (274, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 325, ( :5/2), −0.447+0.894i)(2,\ 325,\ (\ :5/2),\ -0.447 + 0.894i)

Particular Values

L(3)L(3) ≈\approx 0.49333045540.4933304554
L(12)L(\frac12) ≈\approx 0.49333045540.4933304554
L(72)L(\frac{7}{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)
bad5 1 1
13 1+169iT 1 + 169iT
good2 1+1.96iT−32T2 1 + 1.96iT - 32T^{2}
3 1−29.7iT−243T2 1 - 29.7iT - 243T^{2}
7 1−176.iT−1.68e4T2 1 - 176. iT - 1.68e4T^{2}
11 1+62.7T+1.61e5T2 1 + 62.7T + 1.61e5T^{2}
17 1+1.82e3iT−1.41e6T2 1 + 1.82e3iT - 1.41e6T^{2}
19 1+2.50e3T+2.47e6T2 1 + 2.50e3T + 2.47e6T^{2}
23 1+137.iT−6.43e6T2 1 + 137. iT - 6.43e6T^{2}
29 1+5.97e3T+2.05e7T2 1 + 5.97e3T + 2.05e7T^{2}
31 1+5.20e3T+2.86e7T2 1 + 5.20e3T + 2.86e7T^{2}
37 1+4.27e3iT−6.93e7T2 1 + 4.27e3iT - 6.93e7T^{2}
41 1−1.67e4T+1.15e8T2 1 - 1.67e4T + 1.15e8T^{2}
43 1−2.15e4iT−1.47e8T2 1 - 2.15e4iT - 1.47e8T^{2}
47 1−1.67e4iT−2.29e8T2 1 - 1.67e4iT - 2.29e8T^{2}
53 1+2.74e3iT−4.18e8T2 1 + 2.74e3iT - 4.18e8T^{2}
59 1+1.66e4T+7.14e8T2 1 + 1.66e4T + 7.14e8T^{2}
61 1+2.50e4T+8.44e8T2 1 + 2.50e4T + 8.44e8T^{2}
67 1−2.94e4iT−1.35e9T2 1 - 2.94e4iT - 1.35e9T^{2}
71 1+2.69e4T+1.80e9T2 1 + 2.69e4T + 1.80e9T^{2}
73 1+3.36e4iT−2.07e9T2 1 + 3.36e4iT - 2.07e9T^{2}
79 1−1.35e4T+3.07e9T2 1 - 1.35e4T + 3.07e9T^{2}
83 1+790.iT−3.93e9T2 1 + 790. iT - 3.93e9T^{2}
89 1−7.22e4T+5.58e9T2 1 - 7.22e4T + 5.58e9T^{2}
97 1+5.45e3iT−8.58e9T2 1 + 5.45e3iT - 8.58e9T^{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

−11.11541607837833626055802235058, −10.68516924903941286358940554087, −9.431029792333897162473216190423, −9.157216984587382052070805371534, −7.83442040614640277475029111067, −6.14988792432795906345773376101, −5.42844610650417452514001042372, −4.30932565319622453636770288905, −3.05540744040590152329196563285, −2.36525825410572770755879608990, 0.11150717369483002640838670940, 1.48730986556751410052349299993, 2.18357623263459783557937227795, 3.76535894237427656116041944346, 5.75701587428554165323098806944, 6.49930741654576706668309943845, 7.22484367495488484515803069028, 7.81442094704853059464585051372, 8.691649821817345043721489945482, 10.64686780396259115703354517394

Graph of the ZZ-function along the critical line