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Properties

Label 2-15e2-45.4-c3-0-8
Degree $2$
Conductor $225$
Sign $-0.932 - 0.360i$
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 + 1.09i)2-s + (−5.19 − 0.206i)3-s + (−1.59 + 2.75i)4-s + (10.0 − 5.30i)6-s + (2.39 − 1.38i)7-s − 24.5i·8-s + (26.9 + 2.14i)9-s + (26.3 + 45.6i)11-s + (8.83 − 13.9i)12-s + (17.7 + 10.2i)13-s + (−3.03 + 5.25i)14-s + (14.1 + 24.5i)16-s + 3.66i·17-s + (−53.4 + 25.4i)18-s + 95.6·19-s + ⋯
L(s)  = 1  + (−0.671 + 0.387i)2-s + (−0.999 − 0.0396i)3-s + (−0.199 + 0.344i)4-s + (0.686 − 0.360i)6-s + (0.129 − 0.0746i)7-s − 1.08i·8-s + (0.996 + 0.0792i)9-s + (0.721 + 1.25i)11-s + (0.212 − 0.336i)12-s + (0.377 + 0.218i)13-s + (−0.0579 + 0.100i)14-s + (0.221 + 0.384i)16-s + 0.0522i·17-s + (−0.700 + 0.333i)18-s + 1.15·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 - 0.360i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.932 - 0.360i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ -0.932 - 0.360i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0933134 + 0.500735i\)
\(L(\frac12)\) \(\approx\) \(0.0933134 + 0.500735i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (5.19 + 0.206i)T \)
5 \( 1 \)
good2 \( 1 + (1.90 - 1.09i)T + (4 - 6.92i)T^{2} \)
7 \( 1 + (-2.39 + 1.38i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (-26.3 - 45.6i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-17.7 - 10.2i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 3.66iT - 4.91e3T^{2} \)
19 \( 1 - 95.6T + 6.85e3T^{2} \)
23 \( 1 + (77.8 + 44.9i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (113. + 197. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (139. - 241. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 273. iT - 5.06e4T^{2} \)
41 \( 1 + (32.4 - 56.1i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (362. - 209. i)T + (3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-120. + 69.3i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 197. iT - 1.48e5T^{2} \)
59 \( 1 + (-370. + 641. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (244. + 423. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-356. - 205. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 310.T + 3.57e5T^{2} \)
73 \( 1 - 51.0iT - 3.89e5T^{2} \)
79 \( 1 + (-603. - 1.04e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (783. - 452. i)T + (2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 663.T + 7.04e5T^{2} \)
97 \( 1 + (628. - 362. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−12.14893951788543929748449764242, −11.34624775677471249587626080079, −9.985808682052422219785353699104, −9.502348900439939477297992934804, −8.132399509004586869683147828383, −7.15813201111623060637176552898, −6.41073176840742072458345470162, −4.90928135238212279007233092296, −3.83868741517705341863717110157, −1.37935516235518029409895883210, 0.34757878986413229855044082046, 1.54553455964168863404740864402, 3.73330566829132220155747061871, 5.36158748730812242216847954288, 5.92079400289970750611227986361, 7.37904789383202320410570169411, 8.688320739227968215805618026867, 9.535212950155654647762662809238, 10.50839749756287502229882687005, 11.32849416982824761977594357375

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