Properties

Label 2-15e2-25.16-c3-0-34
Degree $2$
Conductor $225$
Sign $0.0234 - 0.999i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.68 − 1.94i)2-s + (0.919 + 2.83i)4-s + (−2.31 − 10.9i)5-s + 10.7·7-s + (−5.14 + 15.8i)8-s + (−15.1 + 33.8i)10-s + (−43.0 − 31.2i)11-s + (−29.8 + 21.6i)13-s + (−28.8 − 20.9i)14-s + (63.8 − 46.4i)16-s + (21.1 − 65.1i)17-s + (−1.49 + 4.60i)19-s + (28.8 − 16.6i)20-s + (54.4 + 167. i)22-s + (−67.0 − 48.7i)23-s + ⋯
L(s)  = 1  + (−0.947 − 0.688i)2-s + (0.114 + 0.353i)4-s + (−0.206 − 0.978i)5-s + 0.581·7-s + (−0.227 + 0.699i)8-s + (−0.477 + 1.06i)10-s + (−1.17 − 0.857i)11-s + (−0.636 + 0.462i)13-s + (−0.551 − 0.400i)14-s + (0.998 − 0.725i)16-s + (0.302 − 0.929i)17-s + (−0.0180 + 0.0555i)19-s + (0.322 − 0.185i)20-s + (0.527 + 1.62i)22-s + (−0.607 − 0.441i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0234 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0234 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.0234 - 0.999i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 0.0234 - 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0161975 + 0.0158224i\)
\(L(\frac12)\) \(\approx\) \(0.0161975 + 0.0158224i\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.31 + 10.9i)T \)
good2 \( 1 + (2.68 + 1.94i)T + (2.47 + 7.60i)T^{2} \)
7 \( 1 - 10.7T + 343T^{2} \)
11 \( 1 + (43.0 + 31.2i)T + (411. + 1.26e3i)T^{2} \)
13 \( 1 + (29.8 - 21.6i)T + (678. - 2.08e3i)T^{2} \)
17 \( 1 + (-21.1 + 65.1i)T + (-3.97e3 - 2.88e3i)T^{2} \)
19 \( 1 + (1.49 - 4.60i)T + (-5.54e3 - 4.03e3i)T^{2} \)
23 \( 1 + (67.0 + 48.7i)T + (3.75e3 + 1.15e4i)T^{2} \)
29 \( 1 + (-51.1 - 157. i)T + (-1.97e4 + 1.43e4i)T^{2} \)
31 \( 1 + (97.3 - 299. i)T + (-2.41e4 - 1.75e4i)T^{2} \)
37 \( 1 + (-202. + 146. i)T + (1.56e4 - 4.81e4i)T^{2} \)
41 \( 1 + (292. - 212. i)T + (2.12e4 - 6.55e4i)T^{2} \)
43 \( 1 - 180.T + 7.95e4T^{2} \)
47 \( 1 + (-68.6 - 211. i)T + (-8.39e4 + 6.10e4i)T^{2} \)
53 \( 1 + (-162. - 499. i)T + (-1.20e5 + 8.75e4i)T^{2} \)
59 \( 1 + (34.9 - 25.3i)T + (6.34e4 - 1.95e5i)T^{2} \)
61 \( 1 + (-502. - 364. i)T + (7.01e4 + 2.15e5i)T^{2} \)
67 \( 1 + (-71.8 + 221. i)T + (-2.43e5 - 1.76e5i)T^{2} \)
71 \( 1 + (-195. - 602. i)T + (-2.89e5 + 2.10e5i)T^{2} \)
73 \( 1 + (770. + 559. i)T + (1.20e5 + 3.69e5i)T^{2} \)
79 \( 1 + (281. + 866. i)T + (-3.98e5 + 2.89e5i)T^{2} \)
83 \( 1 + (-61.3 + 188. i)T + (-4.62e5 - 3.36e5i)T^{2} \)
89 \( 1 + (703. + 511. i)T + (2.17e5 + 6.70e5i)T^{2} \)
97 \( 1 + (68.4 + 210. i)T + (-7.38e5 + 5.36e5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.97872792940734411396159062258, −10.10291858557736294253625930868, −9.059948459338817838904206680108, −8.388771928089266262959530260237, −7.50839853782139985265462428122, −5.55098405983025967991862482327, −4.76338393825376405033849944403, −2.80040544741697345821678644504, −1.32903562434265618392088045638, −0.01336474992095155423387549464, 2.32124832878790747876077904030, 3.94940584065321588544991281098, 5.58472463360696991287625801978, 6.82471816508696910297733266872, 7.83993332082322305719060621223, 8.069596344043330258630986090863, 9.806791379999481464307084491459, 10.17710373955163226203482752229, 11.35485391438968440011411641543, 12.47857825006427572232244915131

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