↑

Properties

Label 2-15e2-15.2-c3-0-12
Degree $2$
Conductor $225$
Sign $0.391 + 0.920i$
Analytic cond. $13.2754$
Root an. cond. $3.64354$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.287 + 0.287i)2-s + 7.83i·4-s + (−15.0 − 15.0i)7-s + (−4.55 − 4.55i)8-s − 27.9i·11-s + (−2.49 + 2.49i)13-s + 8.66·14-s − 60.0·16-s + (67.9 − 67.9i)17-s − 95.2i·19-s + (8.06 + 8.06i)22-s + (121. + 121. i)23-s − 1.43i·26-s + (117. − 117. i)28-s + 99.0·29-s + ⋯
L(s)  = 1  + (−0.101 + 0.101i)2-s + 0.979i·4-s + (−0.812 − 0.812i)7-s + (−0.201 − 0.201i)8-s − 0.767i·11-s + (−0.0533 + 0.0533i)13-s + 0.165·14-s − 0.938·16-s + (0.968 − 0.968i)17-s − 1.14i·19-s + (0.0781 + 0.0781i)22-s + (1.10 + 1.10i)23-s − 0.0108i·26-s + (0.796 − 0.796i)28-s + 0.634·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.391 + 0.920i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 0.391 + 0.920i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.918370 - 0.607647i\)
\(L(\frac12)\) \(\approx\) \(0.918370 - 0.607647i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (0.287 - 0.287i)T - 8iT^{2} \)
7 \( 1 + (15.0 + 15.0i)T + 343iT^{2} \)
11 \( 1 + 27.9iT - 1.33e3T^{2} \)
13 \( 1 + (2.49 - 2.49i)T - 2.19e3iT^{2} \)
17 \( 1 + (-67.9 + 67.9i)T - 4.91e3iT^{2} \)
19 \( 1 + 95.2iT - 6.85e3T^{2} \)
23 \( 1 + (-121. - 121. i)T + 1.21e4iT^{2} \)
29 \( 1 - 99.0T + 2.43e4T^{2} \)
31 \( 1 + 28.7T + 2.97e4T^{2} \)
37 \( 1 + (271. + 271. i)T + 5.06e4iT^{2} \)
41 \( 1 + 453. iT - 6.89e4T^{2} \)
43 \( 1 + (30.5 - 30.5i)T - 7.95e4iT^{2} \)
47 \( 1 + (-254. + 254. i)T - 1.03e5iT^{2} \)
53 \( 1 + (224. + 224. i)T + 1.48e5iT^{2} \)
59 \( 1 + 483.T + 2.05e5T^{2} \)
61 \( 1 + 264.T + 2.26e5T^{2} \)
67 \( 1 + (-498. - 498. i)T + 3.00e5iT^{2} \)
71 \( 1 - 609. iT - 3.57e5T^{2} \)
73 \( 1 + (-74.6 + 74.6i)T - 3.89e5iT^{2} \)
79 \( 1 - 406. iT - 4.93e5T^{2} \)
83 \( 1 + (652. + 652. i)T + 5.71e5iT^{2} \)
89 \( 1 - 139.T + 7.04e5T^{2} \)
97 \( 1 + (557. + 557. i)T + 9.12e5iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.66150258902070261613785725905, −10.72670317664363585664721191726, −9.497660545363711408177615185487, −8.715282769784395275712878706973, −7.37234930800028601656634446337, −6.91951471778545352015438448494, −5.31506851763724350074926191651, −3.77138041382915378785184044420, −2.94861853393238186200285650713, −0.49017048916585940133722742649, 1.47510762155877120049704078552, 2.98649474856961870713391646610, 4.72691268588117642439378927489, 5.88643605584369265323610956577, 6.62260866622428921907306673760, 8.151846270483906502764000788984, 9.277210660646269589296683017633, 10.02770453275789274466071204103, 10.76366932753710878544606359355, 12.19035297462844929669917715267

Graph of the $Z$-function along the critical line