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Properties

Label 2-15e2-9.7-c1-0-7
Degree $2$
Conductor $225$
Sign $0.642 - 0.766i$
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  + (−0.258 + 0.448i)2-s + (1.67 + 0.448i)3-s + (0.866 + 1.5i)4-s + (−0.633 + 0.633i)6-s + (1.67 − 2.89i)7-s − 1.93·8-s + (2.59 + 1.50i)9-s + (−0.633 + 1.09i)11-s + (0.776 + 2.89i)12-s + (−1.22 − 2.12i)13-s + (0.866 + 1.5i)14-s + (−1.23 + 2.13i)16-s − 5.27·17-s + (−1.34 + 0.776i)18-s − 0.732·19-s + ⋯
L(s)  = 1  + (−0.183 + 0.316i)2-s + (0.965 + 0.258i)3-s + (0.433 + 0.750i)4-s + (−0.258 + 0.258i)6-s + (0.632 − 1.09i)7-s − 0.683·8-s + (0.866 + 0.5i)9-s + (−0.191 + 0.331i)11-s + (0.224 + 0.836i)12-s + (−0.339 − 0.588i)13-s + (0.231 + 0.400i)14-s + (−0.308 + 0.533i)16-s − 1.28·17-s + (−0.316 + 0.183i)18-s − 0.167·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.642 - 0.766i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ 0.642 - 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47364 + 0.687170i\)
\(L(\frac12)\) \(\approx\) \(1.47364 + 0.687170i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.67 - 0.448i)T \)
5 \( 1 \)
good2 \( 1 + (0.258 - 0.448i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (-1.67 + 2.89i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.633 - 1.09i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.22 + 2.12i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 5.27T + 17T^{2} \)
19 \( 1 + 0.732T + 19T^{2} \)
23 \( 1 + (-0.258 - 0.448i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.232 + 0.401i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.366 + 0.633i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.24T + 37T^{2} \)
41 \( 1 + (3.86 + 6.69i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.328 - 0.568i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.48 - 2.56i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 1.03T + 53T^{2} \)
59 \( 1 + (4.73 + 8.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.33 + 5.76i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.79 + 6.57i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 14.1T + 71T^{2} \)
73 \( 1 + 8.48T + 73T^{2} \)
79 \( 1 + (3.73 - 6.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.98 - 6.90i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 + (7.58 - 13.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.60275436982112030494420770228, −11.24442145879113405303767683719, −10.44356465835473375702132762903, −9.263506865435701917772093051718, −8.206775558123390181965888961375, −7.58888463628090212911589917252, −6.72434842174280717116373143855, −4.72084020697547006275754002554, −3.66816164342689654726684376235, −2.26767747567647797495396750803, 1.83230626160597893482175243783, 2.74889795243138222271211633042, 4.64730575345348049159149528707, 6.02807022779235154750186742735, 7.09630322300164513359345914552, 8.483654780518624158138953973734, 9.061941721173441441142402476422, 10.06501120318354344520563729136, 11.24486037374905440762430202067, 11.94652336552172954396350449478

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