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Properties

Label 2-15e2-9.4-c3-0-2
Degree $2$
Conductor $225$
Sign $0.964 + 0.264i$
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  + (−1.90 − 3.30i)2-s + (−2.77 − 4.39i)3-s + (−3.28 + 5.68i)4-s + (−9.20 + 17.5i)6-s + (−4.99 − 8.64i)7-s − 5.47·8-s + (−11.5 + 24.4i)9-s + (13.6 + 23.5i)11-s + (34.0 − 1.38i)12-s + (−5.96 + 10.3i)13-s + (−19.0 + 33.0i)14-s + (36.7 + 63.5i)16-s + 69.0·17-s + (102. − 8.37i)18-s − 99.8·19-s + ⋯
L(s)  = 1  + (−0.674 − 1.16i)2-s + (−0.534 − 0.844i)3-s + (−0.410 + 0.710i)4-s + (−0.626 + 1.19i)6-s + (−0.269 − 0.467i)7-s − 0.241·8-s + (−0.427 + 0.903i)9-s + (0.373 + 0.646i)11-s + (0.820 − 0.0333i)12-s + (−0.127 + 0.220i)13-s + (−0.363 + 0.630i)14-s + (0.573 + 0.993i)16-s + 0.985·17-s + (1.34 − 0.109i)18-s − 1.20·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.264i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.964 + 0.264i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.964 + 0.264i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 0.964 + 0.264i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.456411 - 0.0614518i\)
\(L(\frac12)\) \(\approx\) \(0.456411 - 0.0614518i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.77 + 4.39i)T \)
5 \( 1 \)
good2 \( 1 + (1.90 + 3.30i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (4.99 + 8.64i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-13.6 - 23.5i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (5.96 - 10.3i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 69.0T + 4.91e3T^{2} \)
19 \( 1 + 99.8T + 6.85e3T^{2} \)
23 \( 1 + (88.8 - 153. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-62.8 - 108. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-73.5 + 127. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 129.T + 5.06e4T^{2} \)
41 \( 1 + (-151. + 262. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (260. + 451. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-221. - 383. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 209.T + 1.48e5T^{2} \)
59 \( 1 + (334. - 579. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-222. - 385. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (219. - 380. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 284.T + 3.57e5T^{2} \)
73 \( 1 + 80.3T + 3.89e5T^{2} \)
79 \( 1 + (-674. - 1.16e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-597. - 1.03e3i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 107.T + 7.04e5T^{2} \)
97 \( 1 + (-225. - 391. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.90986064580426745693029171064, −10.73448372589931135985023747427, −10.12223640501814099283547016912, −9.035091387265808359762750157511, −7.84290546691064971145735528031, −6.80170822419252213379437082264, −5.63857267730660965245811492986, −3.87492673939070624062382097520, −2.27137213240033835955213067768, −1.16233933767300210605397639190, 0.28997934672244190337297964576, 3.16567845160146567399765920155, 4.73645339332288162407102807997, 6.06384793789034615122805560435, 6.41757393846486894467034446495, 8.050682199002017722456549961634, 8.758531338005078502979164832798, 9.727241327948495234201132186976, 10.55206742446278158872717283974, 11.81848017420270865016006914665

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