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Properties

Label 2-15e2-9.7-c3-0-5
Degree $2$
Conductor $225$
Sign $-0.723 - 0.690i$
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  + (0.236 − 0.410i)2-s + (−2.45 − 4.58i)3-s + (3.88 + 6.73i)4-s + (−2.46 − 0.0785i)6-s + (−4.10 + 7.10i)7-s + 7.47·8-s + (−14.9 + 22.4i)9-s + (−2.63 + 4.56i)11-s + (21.3 − 34.3i)12-s + (−38.5 − 66.7i)13-s + (1.94 + 3.36i)14-s + (−29.3 + 50.7i)16-s − 88.9·17-s + (5.67 + 11.4i)18-s − 91.7·19-s + ⋯
L(s)  = 1  + (0.0837 − 0.145i)2-s + (−0.472 − 0.881i)3-s + (0.485 + 0.841i)4-s + (−0.167 − 0.00534i)6-s + (−0.221 + 0.383i)7-s + 0.330·8-s + (−0.554 + 0.832i)9-s + (−0.0721 + 0.124i)11-s + (0.512 − 0.825i)12-s + (−0.821 − 1.42i)13-s + (0.0371 + 0.0643i)14-s + (−0.458 + 0.793i)16-s − 1.26·17-s + (0.0743 + 0.150i)18-s − 1.10·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.723 - 0.690i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.723 - 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.723 - 0.690i$
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: \((2,\ 225,\ (\ :3/2),\ -0.723 - 0.690i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.138284 + 0.345235i\)
\(L(\frac12)\) \(\approx\) \(0.138284 + 0.345235i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.45 + 4.58i)T \)
5 \( 1 \)
good2 \( 1 + (-0.236 + 0.410i)T + (-4 - 6.92i)T^{2} \)
7 \( 1 + (4.10 - 7.10i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (2.63 - 4.56i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (38.5 + 66.7i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 88.9T + 4.91e3T^{2} \)
19 \( 1 + 91.7T + 6.85e3T^{2} \)
23 \( 1 + (-77.2 - 133. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (84.1 - 145. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (36.3 + 63.0i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 154.T + 5.06e4T^{2} \)
41 \( 1 + (3.97 + 6.88i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-11.6 + 20.0i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (150. - 261. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 344.T + 1.48e5T^{2} \)
59 \( 1 + (125. + 217. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (136. - 236. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (417. + 723. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 351.T + 3.57e5T^{2} \)
73 \( 1 - 522.T + 3.89e5T^{2} \)
79 \( 1 + (-350. + 606. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (120. - 208. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 1.02e3T + 7.04e5T^{2} \)
97 \( 1 + (-194. + 337. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−12.45432799877146409345702572268, −11.26573953841236077207851991440, −10.66726870923134417685490289522, −9.034617510124751991074699685302, −7.936094108192784936705252311142, −7.19845117518723528378516493463, −6.18160491436577460752465078610, −4.92133461424071555740507523990, −3.11980109183871035889349519111, −2.00951688113156152369349483982, 0.14228787808288684142514123467, 2.24538444031778879710323453735, 4.19284624516161212130479040219, 4.98242460768763108056020239409, 6.38323853035644166241646714642, 6.89091499400819411339739024535, 8.781226466921025624418715983695, 9.638037146136694028165761149311, 10.58058651074799547449690969090, 11.16379163763953543304322613062

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