Properties

Label 2-15e2-9.4-c3-0-28
Degree $2$
Conductor $225$
Sign $0.948 + 0.315i$
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  + (1.07 + 1.85i)2-s + (−2.66 − 4.46i)3-s + (1.69 − 2.94i)4-s + (5.43 − 9.73i)6-s + (11.6 + 20.1i)7-s + 24.4·8-s + (−12.8 + 23.7i)9-s + (−6.24 − 10.8i)11-s + (−17.6 + 0.249i)12-s + (25.3 − 43.8i)13-s + (−24.9 + 43.2i)14-s + (12.6 + 21.9i)16-s + 63.6·17-s + (−57.9 + 1.63i)18-s + 5.41·19-s + ⋯
L(s)  = 1  + (0.379 + 0.657i)2-s + (−0.512 − 0.858i)3-s + (0.212 − 0.367i)4-s + (0.369 − 0.662i)6-s + (0.627 + 1.08i)7-s + 1.08·8-s + (−0.475 + 0.879i)9-s + (−0.171 − 0.296i)11-s + (−0.424 + 0.00600i)12-s + (0.539 − 0.935i)13-s + (−0.476 + 0.825i)14-s + (0.197 + 0.342i)16-s + 0.908·17-s + (−0.758 + 0.0214i)18-s + 0.0653·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.315i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.948 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.948 + 0.315i$
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: \((2,\ 225,\ (\ :3/2),\ 0.948 + 0.315i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.22177 - 0.359435i\)
\(L(\frac12)\) \(\approx\) \(2.22177 - 0.359435i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.66 + 4.46i)T \)
5 \( 1 \)
good2 \( 1 + (-1.07 - 1.85i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (-11.6 - 20.1i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (6.24 + 10.8i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-25.3 + 43.8i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 63.6T + 4.91e3T^{2} \)
19 \( 1 - 5.41T + 6.85e3T^{2} \)
23 \( 1 + (-44.2 + 76.6i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (72.2 + 125. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-138. + 240. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 243.T + 5.06e4T^{2} \)
41 \( 1 + (234. - 405. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-249. - 431. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (226. + 391. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 223.T + 1.48e5T^{2} \)
59 \( 1 + (57.9 - 100. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (41.3 + 71.7i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (405. - 701. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 134.T + 3.57e5T^{2} \)
73 \( 1 - 707.T + 3.89e5T^{2} \)
79 \( 1 + (208. + 361. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-461. - 799. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 114.T + 7.04e5T^{2} \)
97 \( 1 + (180. + 313. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.68381719116397045391322512619, −11.10588421426745836167831072978, −9.922535271797209236269019246411, −8.234831817731694838886566225854, −7.75847351300572289674350087388, −6.28541847815552310916020221834, −5.77726748898889121291295667070, −4.85663961485999195239295522058, −2.54721445897063188406019456334, −1.07369170041312594241777947219, 1.37550585672371077827221600372, 3.34629005443114876443047209234, 4.20799584840607027205722080251, 5.15076262544433062799674635459, 6.78867293069390228764451743305, 7.78331337454630207320961112690, 9.145704028964125075086174797220, 10.39406869368736265429779767685, 10.87681774505045802157082696922, 11.71845620794396799107616360530

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