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Properties

Label 2-15e2-45.23-c1-0-4
Degree $2$
Conductor $225$
Sign $-0.689 - 0.724i$
Analytic cond. $1.79663$
Root an. cond. $1.34038$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.312 + 1.16i)2-s + (−0.284 + 1.70i)3-s + (0.469 + 0.270i)4-s + (−1.90 − 0.866i)6-s + (4.36 + 1.17i)7-s + (−2.17 + 2.17i)8-s + (−2.83 − 0.972i)9-s + (2.41 − 1.39i)11-s + (−0.596 + 0.724i)12-s + (−2.02 + 0.541i)13-s + (−2.73 + 4.72i)14-s + (−1.31 − 2.27i)16-s + (−3.73 − 3.73i)17-s + (2.02 − 3.00i)18-s − 1.08i·19-s + ⋯
L(s)  = 1  + (−0.220 + 0.824i)2-s + (−0.164 + 0.986i)3-s + (0.234 + 0.135i)4-s + (−0.777 − 0.353i)6-s + (1.65 + 0.442i)7-s + (−0.767 + 0.767i)8-s + (−0.945 − 0.324i)9-s + (0.728 − 0.420i)11-s + (−0.172 + 0.209i)12-s + (−0.560 + 0.150i)13-s + (−0.729 + 1.26i)14-s + (−0.327 − 0.567i)16-s + (−0.905 − 0.905i)17-s + (0.476 − 0.708i)18-s − 0.247i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.689 - 0.724i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.689 - 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.689 - 0.724i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ -0.689 - 0.724i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.491223 + 1.14589i\)
\(L(\frac12)\) \(\approx\) \(0.491223 + 1.14589i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.284 - 1.70i)T \)
5 \( 1 \)
good2 \( 1 + (0.312 - 1.16i)T + (-1.73 - i)T^{2} \)
7 \( 1 + (-4.36 - 1.17i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-2.41 + 1.39i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.02 - 0.541i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (3.73 + 3.73i)T + 17iT^{2} \)
19 \( 1 + 1.08iT - 19T^{2} \)
23 \( 1 + (0.794 + 2.96i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-1.10 - 1.91i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.769 - 1.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.78 + 2.78i)T - 37iT^{2} \)
41 \( 1 + (-9.22 - 5.32i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.53 + 5.71i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (3.09 - 11.5i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-5.58 + 5.58i)T - 53iT^{2} \)
59 \( 1 + (-0.349 + 0.605i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.95 + 8.58i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.181 + 0.675i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 6.85iT - 71T^{2} \)
73 \( 1 + (-3.01 - 3.01i)T + 73iT^{2} \)
79 \( 1 + (9.52 - 5.49i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.00 - 1.34i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + 3.86T + 89T^{2} \)
97 \( 1 + (8.96 + 2.40i)T + (84.0 + 48.5i)T^{2} \)
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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

−12.23429620319349180519985629525, −11.36483249345196540876342081079, −11.00061731280091914726357852431, −9.323094567376097633927380694680, −8.675808424885858086435700667054, −7.73852263497385113716866846142, −6.47085199489434528947397055450, −5.31985431036527374104991977942, −4.43575579775730515930814369678, −2.55863242360609956138044509446, 1.33252395598536253522444903098, 2.26343751755379245107763560560, 4.23043468885624485867704916924, 5.76079321809933737327633180273, 6.95009417541854956832607779058, 7.84511189798323536106681306924, 8.933110212604596681932150923191, 10.29365071426344624563261114269, 11.19010563612853187922005797488, 11.71467166642790950994513803422

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