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Properties

Label 2-15e2-25.11-c1-0-6
Degree $2$
Conductor $225$
Sign $0.876 + 0.481i$
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  + (0.809 − 0.587i)2-s + (−0.309 + 0.951i)4-s + (−0.690 − 2.12i)5-s + 4.47·7-s + (0.927 + 2.85i)8-s + (−1.80 − 1.31i)10-s + (2.61 − 1.90i)11-s + (−2.73 − 1.98i)13-s + (3.61 − 2.62i)14-s + (0.809 + 0.587i)16-s + (0.881 + 2.71i)17-s + (−1 − 3.07i)19-s + 2.23·20-s + (1 − 3.07i)22-s + (−3.61 + 2.62i)23-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (−0.154 + 0.475i)4-s + (−0.309 − 0.951i)5-s + 1.69·7-s + (0.327 + 1.00i)8-s + (−0.572 − 0.415i)10-s + (0.789 − 0.573i)11-s + (−0.758 − 0.551i)13-s + (0.966 − 0.702i)14-s + (0.202 + 0.146i)16-s + (0.213 + 0.658i)17-s + (−0.229 − 0.706i)19-s + 0.499·20-s + (0.213 − 0.656i)22-s + (−0.754 + 0.548i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 + 0.481i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.876 + 0.481i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ 0.876 + 0.481i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63552 - 0.419930i\)
\(L(\frac12)\) \(\approx\) \(1.63552 - 0.419930i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.690 + 2.12i)T \)
good2 \( 1 + (-0.809 + 0.587i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 - 4.47T + 7T^{2} \)
11 \( 1 + (-2.61 + 1.90i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (2.73 + 1.98i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.881 - 2.71i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1 + 3.07i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (3.61 - 2.62i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (1.35 - 4.16i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.23 - 6.88i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (6.54 + 4.75i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (1.11 + 0.812i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 5.70T + 43T^{2} \)
47 \( 1 + (-1.61 + 4.97i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (0.427 - 1.31i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (3.23 + 2.35i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.5 + 0.363i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-1.61 - 4.97i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-0.236 + 0.726i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (2.5 - 1.81i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.09 + 3.35i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-6.16 + 4.47i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-2.73 + 8.42i)T + (-78.4 - 57.0i)T^{2} \)
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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

−12.05632783391819231813050025346, −11.60235054872620918046424035094, −10.58407970202855177728717922052, −8.860013480804966148367741146245, −8.349707992381493234343423797379, −7.42790405120259009925472471808, −5.39051016927796325163416647458, −4.71824789779198342601064649634, −3.62989396027649384536275826280, −1.72749929965010951357273397756, 1.93949826677312113901839068846, 4.08437446134245513321465711808, 4.84822832189948904650503146037, 6.17268030779864550328744546749, 7.17897739639323865238545785313, 8.058179543206038949223117324633, 9.584383007897789083075582476923, 10.41628445832067448545366468273, 11.54234059217981358406644650788, 12.08851342813807354937723965215

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