Properties

Label 2-15e2-25.21-c3-0-20
Degree $2$
Conductor $225$
Sign $0.995 + 0.0992i$
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  + (−0.856 + 2.63i)2-s + (0.254 + 0.184i)4-s + (−7.75 − 8.04i)5-s − 7.15·7-s + (−18.6 + 13.5i)8-s + (27.8 − 13.5i)10-s + (15.3 − 47.3i)11-s + (16.6 + 51.1i)13-s + (6.13 − 18.8i)14-s + (−18.9 − 58.3i)16-s + (−10.7 + 7.80i)17-s + (124. − 90.3i)19-s + (−0.486 − 3.47i)20-s + (111. + 81.1i)22-s + (66.4 − 204. i)23-s + ⋯
L(s)  = 1  + (−0.302 + 0.932i)2-s + (0.0317 + 0.0230i)4-s + (−0.694 − 0.719i)5-s − 0.386·7-s + (−0.824 + 0.598i)8-s + (0.881 − 0.428i)10-s + (0.421 − 1.29i)11-s + (0.354 + 1.09i)13-s + (0.117 − 0.360i)14-s + (−0.296 − 0.912i)16-s + (−0.153 + 0.111i)17-s + (1.50 − 1.09i)19-s + (−0.00543 − 0.0389i)20-s + (1.08 + 0.786i)22-s + (0.602 − 1.85i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0992i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.995 + 0.0992i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 0.995 + 0.0992i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.17713 - 0.0585579i\)
\(L(\frac12)\) \(\approx\) \(1.17713 - 0.0585579i\)
\(L(\frac{5}{2})\) 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 \( 1 + (7.75 + 8.04i)T \)
good2 \( 1 + (0.856 - 2.63i)T + (-6.47 - 4.70i)T^{2} \)
7 \( 1 + 7.15T + 343T^{2} \)
11 \( 1 + (-15.3 + 47.3i)T + (-1.07e3 - 782. i)T^{2} \)
13 \( 1 + (-16.6 - 51.1i)T + (-1.77e3 + 1.29e3i)T^{2} \)
17 \( 1 + (10.7 - 7.80i)T + (1.51e3 - 4.67e3i)T^{2} \)
19 \( 1 + (-124. + 90.3i)T + (2.11e3 - 6.52e3i)T^{2} \)
23 \( 1 + (-66.4 + 204. i)T + (-9.84e3 - 7.15e3i)T^{2} \)
29 \( 1 + (56.2 + 40.8i)T + (7.53e3 + 2.31e4i)T^{2} \)
31 \( 1 + (-122. + 89.1i)T + (9.20e3 - 2.83e4i)T^{2} \)
37 \( 1 + (92.0 + 283. i)T + (-4.09e4 + 2.97e4i)T^{2} \)
41 \( 1 + (-110. - 341. i)T + (-5.57e4 + 4.05e4i)T^{2} \)
43 \( 1 - 451.T + 7.95e4T^{2} \)
47 \( 1 + (-124. - 90.6i)T + (3.20e4 + 9.87e4i)T^{2} \)
53 \( 1 + (-39.5 - 28.7i)T + (4.60e4 + 1.41e5i)T^{2} \)
59 \( 1 + (9.36 + 28.8i)T + (-1.66e5 + 1.20e5i)T^{2} \)
61 \( 1 + (-175. + 539. i)T + (-1.83e5 - 1.33e5i)T^{2} \)
67 \( 1 + (557. - 404. i)T + (9.29e4 - 2.86e5i)T^{2} \)
71 \( 1 + (684. + 497. i)T + (1.10e5 + 3.40e5i)T^{2} \)
73 \( 1 + (110. - 339. i)T + (-3.14e5 - 2.28e5i)T^{2} \)
79 \( 1 + (158. + 115. i)T + (1.52e5 + 4.68e5i)T^{2} \)
83 \( 1 + (-580. + 421. i)T + (1.76e5 - 5.43e5i)T^{2} \)
89 \( 1 + (-371. + 1.14e3i)T + (-5.70e5 - 4.14e5i)T^{2} \)
97 \( 1 + (1.00e3 + 727. i)T + (2.82e5 + 8.68e5i)T^{2} \)
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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

−11.63143333596692393635139378274, −11.14572102208062200562138196697, −9.199533708634069953638510545179, −8.827768197488056863878396612278, −7.76867584084116337367832562621, −6.76987726159489633621080704093, −5.85170546628543704688021792926, −4.43847740781161492768045930481, −3.00569272612202007273863800252, −0.60785052503721146309770686582, 1.27961670617556655951476233217, 2.92770545213649202831126764494, 3.75531277693555148599301484930, 5.60403648488081522809283611546, 6.93316800358164931348206113392, 7.71521058269666745752764026751, 9.293782837104292592268815902201, 10.05818509289369848487398654792, 10.79400923446614222789351353162, 11.86816421747461517645114489528

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