Properties

Label 2-15e2-25.16-c3-0-7
Degree $2$
Conductor $225$
Sign $-0.990 + 0.134i$
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  + (3.76 + 2.73i)2-s + (4.22 + 12.9i)4-s + (−5.34 + 9.82i)5-s − 26.0·7-s + (−8.14 + 25.0i)8-s + (−46.9 + 22.3i)10-s + (2.03 + 1.47i)11-s + (−32.0 + 23.2i)13-s + (−98.0 − 71.2i)14-s + (−10.7 + 7.84i)16-s + (26.2 − 80.8i)17-s + (−44.1 + 135. i)19-s + (−150. − 27.9i)20-s + (3.61 + 11.1i)22-s + (127. + 92.3i)23-s + ⋯
L(s)  = 1  + (1.33 + 0.967i)2-s + (0.527 + 1.62i)4-s + (−0.477 + 0.878i)5-s − 1.40·7-s + (−0.359 + 1.10i)8-s + (−1.48 + 0.707i)10-s + (0.0557 + 0.0405i)11-s + (−0.683 + 0.496i)13-s + (−1.87 − 1.36i)14-s + (−0.168 + 0.122i)16-s + (0.374 − 1.15i)17-s + (−0.532 + 1.64i)19-s + (−1.67 − 0.312i)20-s + (0.0350 + 0.107i)22-s + (1.15 + 0.837i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.134i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.990 + 0.134i$
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: \((2,\ 225,\ (\ :3/2),\ -0.990 + 0.134i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.143879 - 2.13001i\)
\(L(\frac12)\) \(\approx\) \(0.143879 - 2.13001i\)
\(L(\frac{5}{2})\) 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 \( 1 + (5.34 - 9.82i)T \)
good2 \( 1 + (-3.76 - 2.73i)T + (2.47 + 7.60i)T^{2} \)
7 \( 1 + 26.0T + 343T^{2} \)
11 \( 1 + (-2.03 - 1.47i)T + (411. + 1.26e3i)T^{2} \)
13 \( 1 + (32.0 - 23.2i)T + (678. - 2.08e3i)T^{2} \)
17 \( 1 + (-26.2 + 80.8i)T + (-3.97e3 - 2.88e3i)T^{2} \)
19 \( 1 + (44.1 - 135. i)T + (-5.54e3 - 4.03e3i)T^{2} \)
23 \( 1 + (-127. - 92.3i)T + (3.75e3 + 1.15e4i)T^{2} \)
29 \( 1 + (-30.9 - 95.3i)T + (-1.97e4 + 1.43e4i)T^{2} \)
31 \( 1 + (53.5 - 164. i)T + (-2.41e4 - 1.75e4i)T^{2} \)
37 \( 1 + (-48.0 + 34.9i)T + (1.56e4 - 4.81e4i)T^{2} \)
41 \( 1 + (287. - 208. i)T + (2.12e4 - 6.55e4i)T^{2} \)
43 \( 1 - 109.T + 7.95e4T^{2} \)
47 \( 1 + (-17.9 - 55.2i)T + (-8.39e4 + 6.10e4i)T^{2} \)
53 \( 1 + (-120. - 371. i)T + (-1.20e5 + 8.75e4i)T^{2} \)
59 \( 1 + (-333. + 242. i)T + (6.34e4 - 1.95e5i)T^{2} \)
61 \( 1 + (290. + 210. i)T + (7.01e4 + 2.15e5i)T^{2} \)
67 \( 1 + (-108. + 333. i)T + (-2.43e5 - 1.76e5i)T^{2} \)
71 \( 1 + (52.7 + 162. i)T + (-2.89e5 + 2.10e5i)T^{2} \)
73 \( 1 + (-754. - 548. i)T + (1.20e5 + 3.69e5i)T^{2} \)
79 \( 1 + (405. + 1.24e3i)T + (-3.98e5 + 2.89e5i)T^{2} \)
83 \( 1 + (-202. + 621. i)T + (-4.62e5 - 3.36e5i)T^{2} \)
89 \( 1 + (-857. - 622. i)T + (2.17e5 + 6.70e5i)T^{2} \)
97 \( 1 + (-198. - 610. i)T + (-7.38e5 + 5.36e5i)T^{2} \)
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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

−12.45037533099074824574699959413, −11.83854214234099060094197425789, −10.40783432062149037001779390717, −9.390027602605916733715214890072, −7.71424846367843145743516501902, −6.94412911952356344499819909607, −6.30098025097428303202268665408, −5.07829453762690129069119112227, −3.69398992555173655866878260353, −2.99713579375233251360407797753, 0.55060459015254465766994528209, 2.51944739381819692101130512902, 3.64269159428250190384982956019, 4.63870399913403465995568983486, 5.69503066729667957689426321065, 6.87947687366736973607613340608, 8.491505223805623759524921109534, 9.634077684860090240978843595173, 10.59408287313128618534850988518, 11.59845357578244707236989665793

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