↑

Properties

Label 2-325-5.4-c5-0-43
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  − 10.1i·2-s + 24.5i·3-s − 70.0·4-s + 247.·6-s − 72.1i·7-s + 384. i·8-s − 358.·9-s + 127.·11-s − 1.71e3i·12-s − 169i·13-s − 728.·14-s + 1.64e3·16-s + 2.15e3i·17-s + 3.61e3i·18-s − 2.72e3·19-s + ⋯
L(s)  = 1  − 1.78i·2-s + 1.57i·3-s − 2.18·4-s + 2.80·6-s − 0.556i·7-s + 2.12i·8-s − 1.47·9-s + 0.317·11-s − 3.44i·12-s − 0.277i·13-s − 0.993·14-s + 1.60·16-s + 1.80i·17-s + 2.63i·18-s − 1.73·19-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 1.2262037821.226203782
L(12)L(\frac12) ≈\approx 1.2262037821.226203782
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+10.1iT−32T2 1 + 10.1iT - 32T^{2}
3 1−24.5iT−243T2 1 - 24.5iT - 243T^{2}
7 1+72.1iT−1.68e4T2 1 + 72.1iT - 1.68e4T^{2}
11 1−127.T+1.61e5T2 1 - 127.T + 1.61e5T^{2}
17 1−2.15e3iT−1.41e6T2 1 - 2.15e3iT - 1.41e6T^{2}
19 1+2.72e3T+2.47e6T2 1 + 2.72e3T + 2.47e6T^{2}
23 1+2.65e3iT−6.43e6T2 1 + 2.65e3iT - 6.43e6T^{2}
29 1−5.88e3T+2.05e7T2 1 - 5.88e3T + 2.05e7T^{2}
31 1−1.36e3T+2.86e7T2 1 - 1.36e3T + 2.86e7T^{2}
37 1+481.iT−6.93e7T2 1 + 481. iT - 6.93e7T^{2}
41 1−7.38e3T+1.15e8T2 1 - 7.38e3T + 1.15e8T^{2}
43 1+1.01e4iT−1.47e8T2 1 + 1.01e4iT - 1.47e8T^{2}
47 1+1.63e4iT−2.29e8T2 1 + 1.63e4iT - 2.29e8T^{2}
53 1−9.06e3iT−4.18e8T2 1 - 9.06e3iT - 4.18e8T^{2}
59 1+2.95e4T+7.14e8T2 1 + 2.95e4T + 7.14e8T^{2}
61 1−1.64e4T+8.44e8T2 1 - 1.64e4T + 8.44e8T^{2}
67 1+6.15e4iT−1.35e9T2 1 + 6.15e4iT - 1.35e9T^{2}
71 1+5.22e3T+1.80e9T2 1 + 5.22e3T + 1.80e9T^{2}
73 1+6.78e4iT−2.07e9T2 1 + 6.78e4iT - 2.07e9T^{2}
79 1−8.95e4T+3.07e9T2 1 - 8.95e4T + 3.07e9T^{2}
83 1+7.89e4iT−3.93e9T2 1 + 7.89e4iT - 3.93e9T^{2}
89 1+1.11e5T+5.58e9T2 1 + 1.11e5T + 5.58e9T^{2}
97 1+7.15e4iT−8.58e9T2 1 + 7.15e4iT - 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

−10.57127637115679268052131421827, −10.09778378512772204987001551623, −8.935077243303355664054559841755, −8.397743745190988775626943034507, −6.20197205332137580795549425688, −4.65205638026373899382546098118, −4.16259405974190355586257935713, −3.34818940643426416303641401175, −2.02993396173403012153945068077, −0.43031295940129504376659832031, 0.875135445223271934174466190226, 2.50781147978392259357850902122, 4.51924846111782127497741709882, 5.67984795001947724957616893709, 6.50722936244026230425058337375, 7.07262865687862113074382046392, 7.949067440612872588068863616525, 8.692238740307307371459756765837, 9.554444891715790156695952603413, 11.39866621158612150537494877409

Graph of the ZZ-function along the critical line