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Properties

Label 2-15e2-45.38-c1-0-9
Degree $2$
Conductor $225$
Sign $-0.221 + 0.975i$
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  + (−0.186 + 0.0499i)2-s + (−1.53 + 0.806i)3-s + (−1.69 + 0.981i)4-s + (0.245 − 0.226i)6-s + (−0.632 − 2.35i)7-s + (0.540 − 0.540i)8-s + (1.69 − 2.47i)9-s + (−2.14 − 1.23i)11-s + (1.81 − 2.87i)12-s + (0.422 − 1.57i)13-s + (0.235 + 0.407i)14-s + (1.88 − 3.27i)16-s + (−0.403 − 0.403i)17-s + (−0.193 + 0.545i)18-s − 4.28i·19-s + ⋯
L(s)  = 1  + (−0.131 + 0.0352i)2-s + (−0.885 + 0.465i)3-s + (−0.849 + 0.490i)4-s + (0.100 − 0.0925i)6-s + (−0.238 − 0.891i)7-s + (0.191 − 0.191i)8-s + (0.566 − 0.823i)9-s + (−0.646 − 0.373i)11-s + (0.523 − 0.829i)12-s + (0.117 − 0.436i)13-s + (0.0629 + 0.108i)14-s + (0.472 − 0.818i)16-s + (−0.0979 − 0.0979i)17-s + (−0.0455 + 0.128i)18-s − 0.983i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.221 + 0.975i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (218, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ -0.221 + 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.211551 - 0.264992i\)
\(L(\frac12)\) \(\approx\) \(0.211551 - 0.264992i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.53 - 0.806i)T \)
5 \( 1 \)
good2 \( 1 + (0.186 - 0.0499i)T + (1.73 - i)T^{2} \)
7 \( 1 + (0.632 + 2.35i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (2.14 + 1.23i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.422 + 1.57i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (0.403 + 0.403i)T + 17iT^{2} \)
19 \( 1 + 4.28iT - 19T^{2} \)
23 \( 1 + (6.82 + 1.82i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (3.20 - 5.55i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.97 + 3.41i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.171 + 0.171i)T - 37iT^{2} \)
41 \( 1 + (6.52 - 3.76i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.95 + 1.32i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (2.91 - 0.780i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-6.12 + 6.12i)T - 53iT^{2} \)
59 \( 1 + (2.27 + 3.93i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.235 - 0.408i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.65 + 0.443i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 3.50iT - 71T^{2} \)
73 \( 1 + (6.88 + 6.88i)T + 73iT^{2} \)
79 \( 1 + (6.50 + 3.75i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.85 - 10.6i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 2.90T + 89T^{2} \)
97 \( 1 + (-0.379 - 1.41i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.96044806223906853655532642414, −10.83594419789438325962307825814, −10.14096816473948545076991419212, −9.186701791181029857338898933758, −7.990942256039739802229095414821, −6.92366413735701724438332706208, −5.55612152407473226971483312823, −4.50609172200792112493085742187, −3.47959780635568080742071585520, −0.33014501568879960977751192340, 1.90077207410429166638275044036, 4.22105877215951841581317402673, 5.49630542378951158884078854800, 6.05183782843952699822630132733, 7.55887066280461619501339135985, 8.614641897924473325108032461646, 9.788113770748531389660189646970, 10.46867653841927118410192996254, 11.71189318059482651109261047966, 12.44892451200254194135821351862

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