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Properties

Label 2-15e2-45.34-c3-0-9
Degree $2$
Conductor $225$
Sign $-0.984 - 0.175i$
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  + (1.18 + 0.686i)2-s + (1.18 + 5.05i)3-s + (−3.05 − 5.29i)4-s + (−2.05 + 6.82i)6-s + (4.43 + 2.55i)7-s − 19.3i·8-s + (−24.1 + 12.0i)9-s + (−27.9 + 48.4i)11-s + (23.1 − 21.7i)12-s + (−32.5 + 18.7i)13-s + (3.51 + 6.08i)14-s + (−11.1 + 19.3i)16-s + 23.6i·17-s + (−36.9 − 2.29i)18-s − 39.0·19-s + ⋯
L(s)  = 1  + (0.420 + 0.242i)2-s + (0.228 + 0.973i)3-s + (−0.382 − 0.662i)4-s + (−0.140 + 0.464i)6-s + (0.239 + 0.138i)7-s − 0.856i·8-s + (−0.895 + 0.445i)9-s + (−0.767 + 1.32i)11-s + (0.557 − 0.523i)12-s + (−0.694 + 0.400i)13-s + (0.0670 + 0.116i)14-s + (−0.174 + 0.302i)16-s + 0.337i·17-s + (−0.484 − 0.0301i)18-s − 0.471·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.175i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.984 - 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.984 - 0.175i$
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: \((2,\ 225,\ (\ :3/2),\ -0.984 - 0.175i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0855770 + 0.966265i\)
\(L(\frac12)\) \(\approx\) \(0.0855770 + 0.966265i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.18 - 5.05i)T \)
5 \( 1 \)
good2 \( 1 + (-1.18 - 0.686i)T + (4 + 6.92i)T^{2} \)
7 \( 1 + (-4.43 - 2.55i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (27.9 - 48.4i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (32.5 - 18.7i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 23.6iT - 4.91e3T^{2} \)
19 \( 1 + 39.0T + 6.85e3T^{2} \)
23 \( 1 + (61.5 - 35.5i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (14.1 - 24.5i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (6.44 + 11.1i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 180. iT - 5.06e4T^{2} \)
41 \( 1 + (-107. - 186. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-53.0 - 30.6i)T + (3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-53.5 - 30.9i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 492. iT - 1.48e5T^{2} \)
59 \( 1 + (-394. - 683. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (260. - 451. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-263. + 152. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 270.T + 3.57e5T^{2} \)
73 \( 1 - 925. iT - 3.89e5T^{2} \)
79 \( 1 + (644. - 1.11e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-618. - 356. i)T + (2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 404.T + 7.04e5T^{2} \)
97 \( 1 + (64.9 + 37.5i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−12.37972598717252405529747636761, −11.08914966029171874386342739757, −10.05106122845718521669216998386, −9.650044485829332323973363771275, −8.454302341720361771474908678804, −7.15770628454173823004949514237, −5.71103085865434443060559291759, −4.83569162187607854087100938139, −4.05040801149020426270940075083, −2.20846242913149782441924187656, 0.32025073469425915522942684010, 2.40030772710599988625240425919, 3.39612365248193403095690087861, 4.96534296537346791265022432841, 6.12290166081982174973607940628, 7.59013212081912071440726659575, 8.144737973065532861591360518081, 9.075697020685511020190214868069, 10.67586043438108166684610811611, 11.62794766581986724059373313865

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