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Properties

Label 2-15e2-25.21-c1-0-0
Degree $2$
Conductor $225$
Sign $-0.929 - 0.368i$
Analytic cond. $1.79663$
Root an. cond. $1.34038$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 + 0.951i)2-s + (0.809 + 0.587i)4-s + (−1.80 + 1.31i)5-s − 4.47·7-s + (−2.42 + 1.76i)8-s + (−0.690 − 2.12i)10-s + (0.381 − 1.17i)11-s + (1.73 + 5.34i)13-s + (1.38 − 4.25i)14-s + (−0.309 − 0.951i)16-s + (3.11 − 2.26i)17-s + (−1 + 0.726i)19-s − 2.23·20-s + (1 + 0.726i)22-s + (−1.38 + 4.25i)23-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (0.404 + 0.293i)4-s + (−0.809 + 0.587i)5-s − 1.69·7-s + (−0.858 + 0.623i)8-s + (−0.218 − 0.672i)10-s + (0.115 − 0.354i)11-s + (0.481 + 1.48i)13-s + (0.369 − 1.13i)14-s + (−0.0772 − 0.237i)16-s + (0.756 − 0.549i)17-s + (−0.229 + 0.166i)19-s − 0.499·20-s + (0.213 + 0.154i)22-s + (−0.288 + 0.886i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.929 - 0.368i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ -0.929 - 0.368i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.137234 + 0.719406i\)
\(L(\frac12)\) \(\approx\) \(0.137234 + 0.719406i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.80 - 1.31i)T \)
good2 \( 1 + (0.309 - 0.951i)T + (-1.61 - 1.17i)T^{2} \)
7 \( 1 + 4.47T + 7T^{2} \)
11 \( 1 + (-0.381 + 1.17i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-1.73 - 5.34i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-3.11 + 2.26i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1 - 0.726i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (1.38 - 4.25i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-5.35 - 3.88i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (2.23 - 1.62i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.954 + 2.93i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.11 - 3.44i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 7.70T + 43T^{2} \)
47 \( 1 + (0.618 + 0.449i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.92 - 2.12i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.23 - 3.80i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-0.5 + 1.53i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (0.618 - 0.449i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (4.23 + 3.07i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.5 - 7.69i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-10.0 + 7.33i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (1.66 - 5.11i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (1.73 + 1.26i)T + (29.9 + 92.2i)T^{2} \)
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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

−12.42513397153867607821452129614, −11.81211032026116063037602691493, −10.83017936497734955758417182909, −9.556752124971673540435322945516, −8.661811948890745460984599028936, −7.39078886440707774786923270002, −6.75589357606135995435645406173, −5.95612910094791863362456194340, −3.83061923643396469933616793890, −2.92785447095874564984542966191, 0.63272342484607830176753672313, 2.86701938106776856420203548953, 3.85292289981108673409549077403, 5.70953934062721895829433189962, 6.64369002915204428445898605805, 7.939672098972339638103391939996, 9.107757720387677150498948971439, 10.09557665791268958063615965319, 10.67201126367363744268785477583, 12.01660421218647469972208810487

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