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Properties

Label 2-15e2-9.7-c3-0-51
Degree $2$
Conductor $225$
Sign $-0.493 - 0.869i$
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  + (2.39 − 4.14i)2-s + (−1.75 − 4.88i)3-s + (−7.46 − 12.9i)4-s + (−24.4 − 4.41i)6-s + (15.3 − 26.5i)7-s − 33.1·8-s + (−20.8 + 17.1i)9-s + (−21.3 + 37.0i)11-s + (−50.1 + 59.2i)12-s + (−26.7 − 46.2i)13-s + (−73.4 − 127. i)14-s + (−19.7 + 34.2i)16-s + 0.609·17-s + (21.4 + 127. i)18-s + 94.5·19-s + ⋯
L(s)  = 1  + (0.846 − 1.46i)2-s + (−0.338 − 0.941i)3-s + (−0.933 − 1.61i)4-s + (−1.66 − 0.300i)6-s + (0.828 − 1.43i)7-s − 1.46·8-s + (−0.771 + 0.636i)9-s + (−0.586 + 1.01i)11-s + (−1.20 + 1.42i)12-s + (−0.570 − 0.987i)13-s + (−1.40 − 2.42i)14-s + (−0.308 + 0.534i)16-s + 0.00869·17-s + (0.281 + 1.66i)18-s + 1.14·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.493 - 0.869i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.493 - 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.493 - 0.869i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ -0.493 - 0.869i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.07211 + 1.84043i\)
\(L(\frac12)\) \(\approx\) \(1.07211 + 1.84043i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.75 + 4.88i)T \)
5 \( 1 \)
good2 \( 1 + (-2.39 + 4.14i)T + (-4 - 6.92i)T^{2} \)
7 \( 1 + (-15.3 + 26.5i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (21.3 - 37.0i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (26.7 + 46.2i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 0.609T + 4.91e3T^{2} \)
19 \( 1 - 94.5T + 6.85e3T^{2} \)
23 \( 1 + (-76.3 - 132. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-71.6 + 124. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (18.6 + 32.3i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 274.T + 5.06e4T^{2} \)
41 \( 1 + (-136. - 236. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (92.6 - 160. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-219. + 379. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 211.T + 1.48e5T^{2} \)
59 \( 1 + (248. + 430. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-104. + 181. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (483. + 837. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 401.T + 3.57e5T^{2} \)
73 \( 1 - 97.6T + 3.89e5T^{2} \)
79 \( 1 + (566. - 980. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-286. + 496. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 90.6T + 7.04e5T^{2} \)
97 \( 1 + (-4.62 + 8.00i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.35419459128035190326110580834, −10.53865921299619143810508693105, −9.818435166206320912232763415825, −7.81637752030287208205364014741, −7.25205155453510858454501433353, −5.38460642124170740754106691714, −4.68714450643323500483703313724, −3.18910015020223006389328658757, −1.80591717915197857707219164112, −0.72095419653774454072534841764, 2.97035046134707321596930954404, 4.52284421232525555832975686944, 5.29986358060820949634491494215, 5.90003403394278509465776872013, 7.20256756089981609728042647007, 8.619076452767813443802727323249, 8.928208266119909284341933360374, 10.60908435462872191579315579606, 11.73075932989225824122232546249, 12.41241563602292394403390625943

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