Properties

Label 2-15e2-45.38-c3-0-17
Degree $2$
Conductor $225$
Sign $0.870 - 0.492i$
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  + (2.79 − 0.748i)2-s + (−5.18 + 0.352i)3-s + (0.310 − 0.179i)4-s + (−14.2 + 4.86i)6-s + (−5.03 − 18.7i)7-s + (−15.6 + 15.6i)8-s + (26.7 − 3.65i)9-s + (49.0 + 28.3i)11-s + (−1.54 + 1.03i)12-s + (0.616 − 2.30i)13-s + (−28.1 − 48.6i)14-s + (−33.3 + 57.7i)16-s + (92.5 + 92.5i)17-s + (71.9 − 30.2i)18-s + 21.7i·19-s + ⋯
L(s)  = 1  + (0.987 − 0.264i)2-s + (−0.997 + 0.0678i)3-s + (0.0388 − 0.0224i)4-s + (−0.967 + 0.330i)6-s + (−0.271 − 1.01i)7-s + (−0.690 + 0.690i)8-s + (0.990 − 0.135i)9-s + (1.34 + 0.776i)11-s + (−0.0371 + 0.0249i)12-s + (0.0131 − 0.0490i)13-s + (−0.536 − 0.929i)14-s + (−0.521 + 0.903i)16-s + (1.32 + 1.32i)17-s + (0.942 − 0.395i)18-s + 0.262i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 - 0.492i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.870 - 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.870 - 0.492i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (218, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 0.870 - 0.492i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.83714 + 0.483562i\)
\(L(\frac12)\) \(\approx\) \(1.83714 + 0.483562i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (5.18 - 0.352i)T \)
5 \( 1 \)
good2 \( 1 + (-2.79 + 0.748i)T + (6.92 - 4i)T^{2} \)
7 \( 1 + (5.03 + 18.7i)T + (-297. + 171.5i)T^{2} \)
11 \( 1 + (-49.0 - 28.3i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-0.616 + 2.30i)T + (-1.90e3 - 1.09e3i)T^{2} \)
17 \( 1 + (-92.5 - 92.5i)T + 4.91e3iT^{2} \)
19 \( 1 - 21.7iT - 6.85e3T^{2} \)
23 \( 1 + (-98.8 - 26.4i)T + (1.05e4 + 6.08e3i)T^{2} \)
29 \( 1 + (40.6 - 70.3i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-4.26 - 7.38i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (8.92 - 8.92i)T - 5.06e4iT^{2} \)
41 \( 1 + (28.1 - 16.2i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-428. + 114. i)T + (6.88e4 - 3.97e4i)T^{2} \)
47 \( 1 + (-140. + 37.5i)T + (8.99e4 - 5.19e4i)T^{2} \)
53 \( 1 + (82.9 - 82.9i)T - 1.48e5iT^{2} \)
59 \( 1 + (129. + 224. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (282. - 489. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (434. + 116. i)T + (2.60e5 + 1.50e5i)T^{2} \)
71 \( 1 + 16.2iT - 3.57e5T^{2} \)
73 \( 1 + (59.2 + 59.2i)T + 3.89e5iT^{2} \)
79 \( 1 + (-472. - 272. i)T + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-260. - 972. i)T + (-4.95e5 + 2.85e5i)T^{2} \)
89 \( 1 + 844.T + 7.04e5T^{2} \)
97 \( 1 + (-105. - 393. i)T + (-7.90e5 + 4.56e5i)T^{2} \)
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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

−12.23474847860398105277568957157, −11.10234528119388134039382501909, −10.23732968266080090582627891888, −9.173412282151890840809543158964, −7.54329949238210053787147013203, −6.48950201698515791034416326980, −5.49320464577420818178027997599, −4.27051690150203315557181632738, −3.65449662802699440441879699088, −1.27999199208704480044451175198, 0.795277875141372549674529784146, 3.16840524555209999858058810012, 4.51783863327973643506357883587, 5.58600222987875583909413611043, 6.14473098426829497259508646287, 7.20888936940061773473210903482, 9.029675661009659202173652320257, 9.659905675269648259232440936837, 11.17488268359275314238416376656, 12.00163051520684157322608370143

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