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Properties

Label 2-15e2-5.2-c4-0-5
Degree $2$
Conductor $225$
Sign $0.525 - 0.850i$
Analytic cond. $23.2582$
Root an. cond. $4.82267$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.02 − 5.02i)2-s + 34.5i·4-s + (38.5 + 38.5i)7-s + (93.3 − 93.3i)8-s + 40.4·11-s + (−20.8 + 20.8i)13-s − 387. i·14-s − 385.·16-s + (15.8 + 15.8i)17-s − 314. i·19-s + (−203. − 203. i)22-s + (−572. + 572. i)23-s + 209.·26-s + (−1.33e3 + 1.33e3i)28-s + 824. i·29-s + ⋯
L(s)  = 1  + (−1.25 − 1.25i)2-s + 2.16i·4-s + (0.786 + 0.786i)7-s + (1.45 − 1.45i)8-s + 0.334·11-s + (−0.123 + 0.123i)13-s − 1.97i·14-s − 1.50·16-s + (0.0549 + 0.0549i)17-s − 0.872i·19-s + (−0.420 − 0.420i)22-s + (−1.08 + 1.08i)23-s + 0.310·26-s + (−1.69 + 1.69i)28-s + 0.980i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.525 - 0.850i$
Analytic conductor: \(23.2582\)
Root analytic conductor: \(4.82267\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :2),\ 0.525 - 0.850i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.5622742221\)
\(L(\frac12)\) \(\approx\) \(0.5622742221\)
\(L(3)\) 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 \( 1 \)
good2 \( 1 + (5.02 + 5.02i)T + 16iT^{2} \)
7 \( 1 + (-38.5 - 38.5i)T + 2.40e3iT^{2} \)
11 \( 1 - 40.4T + 1.46e4T^{2} \)
13 \( 1 + (20.8 - 20.8i)T - 2.85e4iT^{2} \)
17 \( 1 + (-15.8 - 15.8i)T + 8.35e4iT^{2} \)
19 \( 1 + 314. iT - 1.30e5T^{2} \)
23 \( 1 + (572. - 572. i)T - 2.79e5iT^{2} \)
29 \( 1 - 824. iT - 7.07e5T^{2} \)
31 \( 1 + 1.34e3T + 9.23e5T^{2} \)
37 \( 1 + (-589. - 589. i)T + 1.87e6iT^{2} \)
41 \( 1 + 1.85e3T + 2.82e6T^{2} \)
43 \( 1 + (-671. + 671. i)T - 3.41e6iT^{2} \)
47 \( 1 + (-504. - 504. i)T + 4.87e6iT^{2} \)
53 \( 1 + (-2.25e3 + 2.25e3i)T - 7.89e6iT^{2} \)
59 \( 1 - 2.58e3iT - 1.21e7T^{2} \)
61 \( 1 + 3.27e3T + 1.38e7T^{2} \)
67 \( 1 + (-3.42e3 - 3.42e3i)T + 2.01e7iT^{2} \)
71 \( 1 - 5.67e3T + 2.54e7T^{2} \)
73 \( 1 + (4.45e3 - 4.45e3i)T - 2.83e7iT^{2} \)
79 \( 1 - 6.46e3iT - 3.89e7T^{2} \)
83 \( 1 + (621. - 621. i)T - 4.74e7iT^{2} \)
89 \( 1 - 1.85e3iT - 6.27e7T^{2} \)
97 \( 1 + (1.23e4 + 1.23e4i)T + 8.85e7iT^{2} \)
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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

−11.57916956370932287923625589864, −10.84456965869280605104392225232, −9.750124312283163827178585095933, −8.967615728783243810166495996263, −8.225391264833742925092134939627, −7.14671113807697778849672244562, −5.36751254430701154564900412398, −3.76725112811468139525477237181, −2.39285339068127952857866406431, −1.39017592600649383431813536138, 0.30776142888827612518196722613, 1.67605597084748253708281215200, 4.19695697819104388272977912590, 5.57674313054964358793257856503, 6.59222317993751631108681265450, 7.65428505600687847327854605178, 8.175095695044027853713660105879, 9.289524895719231463102122211301, 10.22174905199318148063098027353, 10.94878249565967853244039540801

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