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Properties

Label 2-15e2-45.34-c1-0-2
Degree $2$
Conductor $225$
Sign $0.537 + 0.843i$
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  + (−1.80 − 1.04i)2-s + (−1.38 − 1.04i)3-s + (1.17 + 2.03i)4-s + (1.41 + 3.32i)6-s + (3.53 + 2.04i)7-s − 0.734i·8-s + (0.824 + 2.88i)9-s + (0.675 − 1.17i)11-s + (0.498 − 4.04i)12-s + (0.561 − 0.324i)13-s + (−4.26 − 7.38i)14-s + (1.58 − 2.74i)16-s + 1.35i·17-s + (1.52 − 6.07i)18-s − 0.648·19-s + ⋯
L(s)  = 1  + (−1.27 − 0.737i)2-s + (−0.798 − 0.602i)3-s + (0.587 + 1.01i)4-s + (0.575 + 1.35i)6-s + (1.33 + 0.772i)7-s − 0.259i·8-s + (0.274 + 0.961i)9-s + (0.203 − 0.353i)11-s + (0.143 − 1.16i)12-s + (0.155 − 0.0898i)13-s + (−1.13 − 1.97i)14-s + (0.396 − 0.686i)16-s + 0.327i·17-s + (0.358 − 1.43i)18-s − 0.148·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 + 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.537 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.537 + 0.843i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ 0.537 + 0.843i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.511051 - 0.280327i\)
\(L(\frac12)\) \(\approx\) \(0.511051 - 0.280327i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.38 + 1.04i)T \)
5 \( 1 \)
good2 \( 1 + (1.80 + 1.04i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (-3.53 - 2.04i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.675 + 1.17i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.561 + 0.324i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.35iT - 17T^{2} \)
19 \( 1 + 0.648T + 19T^{2} \)
23 \( 1 + (-4.14 + 2.39i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.93 + 3.35i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.84 - 6.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 7.52iT - 37T^{2} \)
41 \( 1 + (-0.0898 - 0.155i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.710 - 0.410i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-9.44 - 5.45i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 4.17iT - 53T^{2} \)
59 \( 1 + (-2.08 - 3.61i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.91 + 3.30i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.05 - 4.07i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.11T + 71T^{2} \)
73 \( 1 + 12.3iT - 73T^{2} \)
79 \( 1 + (-5.17 + 8.95i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-10.6 - 6.12i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 + (11.7 + 6.79i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.86105203919390780431922395026, −10.98528732115541454852234988609, −10.53192752922117181547540620384, −9.026235502173736323154219020857, −8.332896702272578135636321187360, −7.43629262203814156723329547776, −5.96005188929348773041424794757, −4.83601824598924290861124958903, −2.41994403340668963101620121779, −1.18363938648143425188446841993, 1.11819197348752860652791839234, 4.11759900805592214076556964362, 5.22657481184928382218971053782, 6.60255303293165370091145481682, 7.44111572248816997963464676826, 8.460388406376987690817513648688, 9.470882409407328664081985254042, 10.35882581327029742990669379077, 11.08492388626079892465020991699, 11.96088313272070035416570853221

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