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Properties

Label 2-15e2-5.4-c3-0-16
Degree $2$
Conductor $225$
Sign $0.894 + 0.447i$
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  + 3.35i·2-s − 3.28·4-s − 30.4i·7-s + 15.8i·8-s − 31.4·11-s − 60.7i·13-s + 102.·14-s − 79.4·16-s − 121. i·17-s + 14.4·19-s − 105. i·22-s − 13.6i·23-s + 204.·26-s + 99.8i·28-s − 76.0·29-s + ⋯
L(s)  = 1  + 1.18i·2-s − 0.410·4-s − 1.64i·7-s + 0.700i·8-s − 0.861·11-s − 1.29i·13-s + 1.95·14-s − 1.24·16-s − 1.72i·17-s + 0.174·19-s − 1.02i·22-s − 0.124i·23-s + 1.53·26-s + 0.674i·28-s − 0.486·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.30848 - 0.308890i\)
\(L(\frac12)\) \(\approx\) \(1.30848 - 0.308890i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 - 3.35iT - 8T^{2} \)
7 \( 1 + 30.4iT - 343T^{2} \)
11 \( 1 + 31.4T + 1.33e3T^{2} \)
13 \( 1 + 60.7iT - 2.19e3T^{2} \)
17 \( 1 + 121. iT - 4.91e3T^{2} \)
19 \( 1 - 14.4T + 6.85e3T^{2} \)
23 \( 1 + 13.6iT - 1.21e4T^{2} \)
29 \( 1 + 76.0T + 2.43e4T^{2} \)
31 \( 1 - 183.T + 2.97e4T^{2} \)
37 \( 1 - 37.3iT - 5.06e4T^{2} \)
41 \( 1 - 30.6T + 6.89e4T^{2} \)
43 \( 1 - 327. iT - 7.95e4T^{2} \)
47 \( 1 + 449. iT - 1.03e5T^{2} \)
53 \( 1 - 301. iT - 1.48e5T^{2} \)
59 \( 1 - 340.T + 2.05e5T^{2} \)
61 \( 1 - 619.T + 2.26e5T^{2} \)
67 \( 1 + 256. iT - 3.00e5T^{2} \)
71 \( 1 + 499.T + 3.57e5T^{2} \)
73 \( 1 + 19.1iT - 3.89e5T^{2} \)
79 \( 1 + 257.T + 4.93e5T^{2} \)
83 \( 1 + 914. iT - 5.71e5T^{2} \)
89 \( 1 + 1.05e3T + 7.04e5T^{2} \)
97 \( 1 + 521iT - 9.12e5T^{2} \)
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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

−11.59434774342559971271700950580, −10.64581476220005459920139836802, −9.815990756044417380716382850112, −8.270650458988506286305172744079, −7.52606291228204270864877318816, −6.92004463861838907403008041955, −5.56205774342940586072239893771, −4.62466622537874793404948940439, −2.92761588223022022066699802169, −0.53703609271847941912098661968, 1.77798480138289847955014572081, 2.65802551571146374984710286764, 4.06285176806909083326625173112, 5.54613981304176158775674222393, 6.63826693073512801114646514262, 8.248670737294874924970049471491, 9.145932203287424892025520314590, 10.06791257841804809760770715401, 11.08027029048880620325261126447, 11.85940772287929258466956524611

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