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Properties

Label 2-15e2-45.32-c3-0-27
Degree $2$
Conductor $225$
Sign $-0.413 - 0.910i$
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  + (5.27 + 1.41i)2-s + (−1.94 + 4.81i)3-s + (18.9 + 10.9i)4-s + (−17.0 + 22.6i)6-s + (−1.13 + 4.22i)7-s + (53.3 + 53.3i)8-s + (−19.4 − 18.7i)9-s + (−5.91 + 3.41i)11-s + (−89.2 + 69.9i)12-s + (6.54 + 24.4i)13-s + (−11.9 + 20.6i)14-s + (118. + 205. i)16-s + (36.0 − 36.0i)17-s + (−76.2 − 126. i)18-s − 59.8i·19-s + ⋯
L(s)  = 1  + (1.86 + 0.499i)2-s + (−0.373 + 0.927i)3-s + (2.36 + 1.36i)4-s + (−1.16 + 1.54i)6-s + (−0.0610 + 0.227i)7-s + (2.35 + 2.35i)8-s + (−0.720 − 0.693i)9-s + (−0.162 + 0.0936i)11-s + (−2.14 + 1.68i)12-s + (0.139 + 0.521i)13-s + (−0.227 + 0.394i)14-s + (1.85 + 3.21i)16-s + (0.514 − 0.514i)17-s + (−0.998 − 1.65i)18-s − 0.722i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.413 - 0.910i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.413 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.413 - 0.910i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ -0.413 - 0.910i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.57345 + 3.99385i\)
\(L(\frac12)\) \(\approx\) \(2.57345 + 3.99385i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.94 - 4.81i)T \)
5 \( 1 \)
good2 \( 1 + (-5.27 - 1.41i)T + (6.92 + 4i)T^{2} \)
7 \( 1 + (1.13 - 4.22i)T + (-297. - 171.5i)T^{2} \)
11 \( 1 + (5.91 - 3.41i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-6.54 - 24.4i)T + (-1.90e3 + 1.09e3i)T^{2} \)
17 \( 1 + (-36.0 + 36.0i)T - 4.91e3iT^{2} \)
19 \( 1 + 59.8iT - 6.85e3T^{2} \)
23 \( 1 + (50.9 - 13.6i)T + (1.05e4 - 6.08e3i)T^{2} \)
29 \( 1 + (-22.1 - 38.2i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-67.9 + 117. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (233. + 233. i)T + 5.06e4iT^{2} \)
41 \( 1 + (-335. - 193. i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-306. - 82.0i)T + (6.88e4 + 3.97e4i)T^{2} \)
47 \( 1 + (316. + 84.9i)T + (8.99e4 + 5.19e4i)T^{2} \)
53 \( 1 + (-43.1 - 43.1i)T + 1.48e5iT^{2} \)
59 \( 1 + (-248. + 431. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-36.3 - 62.8i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (178. - 47.7i)T + (2.60e5 - 1.50e5i)T^{2} \)
71 \( 1 + 563. iT - 3.57e5T^{2} \)
73 \( 1 + (84.7 - 84.7i)T - 3.89e5iT^{2} \)
79 \( 1 + (-186. + 107. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-249. + 930. i)T + (-4.95e5 - 2.85e5i)T^{2} \)
89 \( 1 - 288.T + 7.04e5T^{2} \)
97 \( 1 + (-425. + 1.58e3i)T + (-7.90e5 - 4.56e5i)T^{2} \)
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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

−12.13687005974716713827219597489, −11.49569124054960719854732447236, −10.61428401619661006072025941152, −9.207316985977335835201741745779, −7.71599867164261572705955020676, −6.53092852180895125054487860909, −5.63730207952732643572569113984, −4.75220970000710478875529439505, −3.82571653043453079126977032087, −2.63620490033525462893461892633, 1.26289292705333404837158283356, 2.61606618871291797345582249111, 3.87521483827730432081301078124, 5.27781512847904089615676261564, 6.00545555041887084675210801331, 6.96807525606258795373391082253, 8.059749844053269455986107165195, 10.24400501175143495550802029611, 10.88260964617621685774943339772, 12.01166677634534374427811695095

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