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Properties

Label 2-15e2-25.21-c1-0-3
Degree $2$
Conductor $225$
Sign $-0.197 - 0.980i$
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  + (−0.655 + 2.01i)2-s + (−2.02 − 1.47i)4-s + (2.21 − 0.291i)5-s + 4.35·7-s + (0.865 − 0.629i)8-s + (−0.865 + 4.66i)10-s + (0.488 − 1.50i)11-s + (0.370 + 1.13i)13-s + (−2.85 + 8.79i)14-s + (−0.845 − 2.60i)16-s + (−0.907 + 0.659i)17-s + (−6.21 + 4.51i)19-s + (−4.92 − 2.67i)20-s + (2.71 + 1.97i)22-s + (0.717 − 2.20i)23-s + ⋯
L(s)  = 1  + (−0.463 + 1.42i)2-s + (−1.01 − 0.735i)4-s + (0.991 − 0.130i)5-s + 1.64·7-s + (0.306 − 0.222i)8-s + (−0.273 + 1.47i)10-s + (0.147 − 0.453i)11-s + (0.102 + 0.315i)13-s + (−0.763 + 2.35i)14-s + (−0.211 − 0.650i)16-s + (−0.220 + 0.159i)17-s + (−1.42 + 1.03i)19-s + (−1.10 − 0.597i)20-s + (0.578 + 0.420i)22-s + (0.149 − 0.460i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.197 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.197 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.197 - 0.980i$
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: \((2,\ 225,\ (\ :1/2),\ -0.197 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.768693 + 0.938727i\)
\(L(\frac12)\) \(\approx\) \(0.768693 + 0.938727i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.21 + 0.291i)T \)
good2 \( 1 + (0.655 - 2.01i)T + (-1.61 - 1.17i)T^{2} \)
7 \( 1 - 4.35T + 7T^{2} \)
11 \( 1 + (-0.488 + 1.50i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.370 - 1.13i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.907 - 0.659i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (6.21 - 4.51i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-0.717 + 2.20i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (4.45 + 3.23i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (3.88 - 2.82i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.96 + 6.06i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.30 - 7.10i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 1.24T + 43T^{2} \)
47 \( 1 + (-3.33 - 2.42i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-3.03 - 2.20i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.82 + 8.70i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-0.431 + 1.32i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-3.12 + 2.27i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (8.57 + 6.23i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.54 + 4.75i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (11.7 + 8.55i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (7.06 - 5.13i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-3.10 + 9.54i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-6.06 - 4.40i)T + (29.9 + 92.2i)T^{2} \)
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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

−12.71498066879272713793350954205, −11.36612283510129675133847686048, −10.44685741028517476129650832364, −9.104328222772360258007932743116, −8.488326306150656106969465581763, −7.61767316965794148458912168461, −6.35834167859752808371159767226, −5.60144617304737061449199369029, −4.51096428242402816551181064163, −1.88897830038991048164882198480, 1.55833299302875315169646102304, 2.46344087447409336852857776269, 4.25892552962089246643049440719, 5.46592068175269420521285820341, 7.04879285177182101009301718206, 8.527688657631010017098964253281, 9.181781103400232794587354736956, 10.33742930274827791732954152076, 10.94594285831537616208409962295, 11.65205146869506290378089414104

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