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Properties

Label 2-15e2-45.34-c3-0-47
Degree $2$
Conductor $225$
Sign $0.488 + 0.872i$
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  + (3.24 + 1.87i)2-s + (3.24 − 4.05i)3-s + (3.02 + 5.24i)4-s + (18.1 − 7.08i)6-s + (−27.1 − 15.6i)7-s − 7.30i·8-s + (−5.92 − 26.3i)9-s + (10.4 − 18.0i)11-s + (31.0 + 4.73i)12-s + (51.9 − 29.9i)13-s + (−58.7 − 101. i)14-s + (37.8 − 65.6i)16-s + 74.0i·17-s + (30.1 − 96.6i)18-s + 63.8·19-s + ⋯
L(s)  = 1  + (1.14 + 0.662i)2-s + (0.624 − 0.780i)3-s + (0.378 + 0.655i)4-s + (1.23 − 0.482i)6-s + (−1.46 − 0.846i)7-s − 0.322i·8-s + (−0.219 − 0.975i)9-s + (0.285 − 0.494i)11-s + (0.747 + 0.113i)12-s + (1.10 − 0.639i)13-s + (−1.12 − 1.94i)14-s + (0.592 − 1.02i)16-s + 1.05i·17-s + (0.394 − 1.26i)18-s + 0.770·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.488 + 0.872i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.488 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.488 + 0.872i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 0.488 + 0.872i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.85021 - 1.67101i\)
\(L(\frac12)\) \(\approx\) \(2.85021 - 1.67101i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.24 + 4.05i)T \)
5 \( 1 \)
good2 \( 1 + (-3.24 - 1.87i)T + (4 + 6.92i)T^{2} \)
7 \( 1 + (27.1 + 15.6i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (-10.4 + 18.0i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-51.9 + 29.9i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 74.0iT - 4.91e3T^{2} \)
19 \( 1 - 63.8T + 6.85e3T^{2} \)
23 \( 1 + (28.4 - 16.4i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (80.0 - 138. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-127. - 220. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 215. iT - 5.06e4T^{2} \)
41 \( 1 + (70.8 + 122. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (119. + 68.9i)T + (3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-29.0 - 16.7i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 41.9iT - 1.48e5T^{2} \)
59 \( 1 + (-307. - 532. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-67.1 + 116. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-742. + 428. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 588.T + 3.57e5T^{2} \)
73 \( 1 + 618. iT - 3.89e5T^{2} \)
79 \( 1 + (172. - 299. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-946. - 546. i)T + (2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 414.T + 7.04e5T^{2} \)
97 \( 1 + (174. + 100. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−12.30987674945783575116482976705, −10.69083716076022202393638309827, −9.589681163510907345406961299479, −8.407266567742919126329677338289, −7.17971723706127464320065304141, −6.48714148016926935210129825945, −5.69471840918519181138030039085, −3.69721040598352272707578091374, −3.37205702275415889461564741219, −0.916483328219729323218030929727, 2.35444409267646759148766014905, 3.28812653351995441040447195374, 4.18239700667302371425395848513, 5.40554606612232734118642090026, 6.49289112480628210316384237648, 8.229882387616263227510957555525, 9.374128368176639084381658783535, 9.890137842236870369340406188124, 11.36119503859183179202169705948, 11.92040727088910850162338867118

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