↑

Properties

Label 2-725-145.133-c1-0-24
Degree 22
Conductor 725725
Sign −0.939−0.341i-0.939 - 0.341i
Analytic cond. 5.789155.78915
Root an. cond. 2.406062.40606
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  − 2.32i·2-s − 3.11·3-s − 3.39·4-s + 7.22i·6-s + (2.50 − 2.50i)7-s + 3.23i·8-s + 6.68·9-s + (2.71 − 2.71i)11-s + 10.5·12-s + (2.88 − 2.88i)13-s + (−5.81 − 5.81i)14-s + 0.734·16-s − 4.08i·17-s − 15.5i·18-s + (3.71 + 3.71i)19-s + ⋯
L(s)  = 1  − 1.64i·2-s − 1.79·3-s − 1.69·4-s + 2.95i·6-s + (0.945 − 0.945i)7-s + 1.14i·8-s + 2.22·9-s + (0.817 − 0.817i)11-s + 3.05·12-s + (0.798 − 0.798i)13-s + (−1.55 − 1.55i)14-s + 0.183·16-s − 0.991i·17-s − 3.66i·18-s + (0.851 + 0.851i)19-s + ⋯

Functional equation

Λ(s)=(725s/2 ΓC(s) L(s)=((−0.939−0.341i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.341i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(725s/2 ΓC(s+1/2) L(s)=((−0.939−0.341i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 725725    =    52⋅295^{2} \cdot 29
Sign: −0.939−0.341i-0.939 - 0.341i
Analytic conductor: 5.789155.78915
Root analytic conductor: 2.406062.40606
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ725(568,⋅)\chi_{725} (568, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 725, ( :1/2), −0.939−0.341i)(2,\ 725,\ (\ :1/2),\ -0.939 - 0.341i)

Particular Values

L(1)L(1) ≈\approx 0.160255+0.910282i0.160255 + 0.910282i
L(12)L(\frac12) ≈\approx 0.160255+0.910282i0.160255 + 0.910282i
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)
bad5 1 1
29 1+(−3.51+4.07i)T 1 + (-3.51 + 4.07i)T
good2 1+2.32iT−2T2 1 + 2.32iT - 2T^{2}
3 1+3.11T+3T2 1 + 3.11T + 3T^{2}
7 1+(−2.50+2.50i)T−7iT2 1 + (-2.50 + 2.50i)T - 7iT^{2}
11 1+(−2.71+2.71i)T−11iT2 1 + (-2.71 + 2.71i)T - 11iT^{2}
13 1+(−2.88+2.88i)T−13iT2 1 + (-2.88 + 2.88i)T - 13iT^{2}
17 1+4.08iT−17T2 1 + 4.08iT - 17T^{2}
19 1+(−3.71−3.71i)T+19iT2 1 + (-3.71 - 3.71i)T + 19iT^{2}
23 1+(−2.82−2.82i)T+23iT2 1 + (-2.82 - 2.82i)T + 23iT^{2}
31 1+(−1.31+1.31i)T−31iT2 1 + (-1.31 + 1.31i)T - 31iT^{2}
37 1−7.39T+37T2 1 - 7.39T + 37T^{2}
41 1+(1.29+1.29i)T+41iT2 1 + (1.29 + 1.29i)T + 41iT^{2}
43 1+6.79T+43T2 1 + 6.79T + 43T^{2}
47 1+12.0T+47T2 1 + 12.0T + 47T^{2}
53 1+(−3.85−3.85i)T+53iT2 1 + (-3.85 - 3.85i)T + 53iT^{2}
59 1+0.511iT−59T2 1 + 0.511iT - 59T^{2}
61 1+(−4.83+4.83i)T−61iT2 1 + (-4.83 + 4.83i)T - 61iT^{2}
67 1+(−5.02−5.02i)T+67iT2 1 + (-5.02 - 5.02i)T + 67iT^{2}
71 1−6.77iT−71T2 1 - 6.77iT - 71T^{2}
73 1−5.31iT−73T2 1 - 5.31iT - 73T^{2}
79 1+(8.66+8.66i)T+79iT2 1 + (8.66 + 8.66i)T + 79iT^{2}
83 1+(8.31+8.31i)T+83iT2 1 + (8.31 + 8.31i)T + 83iT^{2}
89 1+(4.54+4.54i)T+89iT2 1 + (4.54 + 4.54i)T + 89iT^{2}
97 1+13.1T+97T2 1 + 13.1T + 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

−10.20823812997379785265924128451, −9.773737316153786317473637564594, −8.363770627148565789752562472010, −7.21943726552328666623301938951, −6.08910496797383430868775060384, −5.13179573203666084575249409245, −4.31230878001516184282380482180, −3.36733718386852788580156100215, −1.30535133733217813200189201215, −0.797908907829593715444239254344, 1.47034089576459317219466999155, 4.34049948599834427583192613746, 4.94153785646360319090891099089, 5.63528604408041664788519143900, 6.66453337542988402591095728223, 6.76598339062572466476955235970, 8.124917966005659538497544067382, 8.920091058327962105248084705100, 9.873011368083252466584519849674, 11.18088986614711944645040563338

Graph of the ZZ-function along the critical line