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Properties

Label 2-15e2-15.8-c3-0-3
Degree $2$
Conductor $225$
Sign $-0.999 - 0.0387i$
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  + (2.80 + 2.80i)2-s + 7.70i·4-s + (−25.0 + 25.0i)7-s + (0.817 − 0.817i)8-s + 42.1i·11-s + (−25.7 − 25.7i)13-s − 140.·14-s + 66.2·16-s + (−42.0 − 42.0i)17-s + 58.9i·19-s + (−118. + 118. i)22-s + (−141. + 141. i)23-s − 144. i·26-s + (−192. − 192. i)28-s − 79.0·29-s + ⋯
L(s)  = 1  + (0.990 + 0.990i)2-s + 0.963i·4-s + (−1.35 + 1.35i)7-s + (0.0361 − 0.0361i)8-s + 1.15i·11-s + (−0.548 − 0.548i)13-s − 2.67·14-s + 1.03·16-s + (−0.600 − 0.600i)17-s + 0.711i·19-s + (−1.14 + 1.14i)22-s + (−1.28 + 1.28i)23-s − 1.08i·26-s + (−1.30 − 1.30i)28-s − 0.506·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0387i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.999 - 0.0387i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ -0.999 - 0.0387i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0366361 + 1.88950i\)
\(L(\frac12)\) \(\approx\) \(0.0366361 + 1.88950i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (-2.80 - 2.80i)T + 8iT^{2} \)
7 \( 1 + (25.0 - 25.0i)T - 343iT^{2} \)
11 \( 1 - 42.1iT - 1.33e3T^{2} \)
13 \( 1 + (25.7 + 25.7i)T + 2.19e3iT^{2} \)
17 \( 1 + (42.0 + 42.0i)T + 4.91e3iT^{2} \)
19 \( 1 - 58.9iT - 6.85e3T^{2} \)
23 \( 1 + (141. - 141. i)T - 1.21e4iT^{2} \)
29 \( 1 + 79.0T + 2.43e4T^{2} \)
31 \( 1 - 184.T + 2.97e4T^{2} \)
37 \( 1 + (-81.4 + 81.4i)T - 5.06e4iT^{2} \)
41 \( 1 - 14.0iT - 6.89e4T^{2} \)
43 \( 1 + (-152. - 152. i)T + 7.95e4iT^{2} \)
47 \( 1 + (-199. - 199. i)T + 1.03e5iT^{2} \)
53 \( 1 + (84.5 - 84.5i)T - 1.48e5iT^{2} \)
59 \( 1 - 665.T + 2.05e5T^{2} \)
61 \( 1 - 600.T + 2.26e5T^{2} \)
67 \( 1 + (-6.22 + 6.22i)T - 3.00e5iT^{2} \)
71 \( 1 + 750. iT - 3.57e5T^{2} \)
73 \( 1 + (-418. - 418. i)T + 3.89e5iT^{2} \)
79 \( 1 - 825. iT - 4.93e5T^{2} \)
83 \( 1 + (463. - 463. i)T - 5.71e5iT^{2} \)
89 \( 1 + 646.T + 7.04e5T^{2} \)
97 \( 1 + (-371. + 371. i)T - 9.12e5iT^{2} \)
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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

−12.54599139286584226254738063259, −11.85584275648150721879457352313, −9.984253956685009592396011716830, −9.479153296546108097162492312129, −7.959097014746970917872305582359, −6.93872907987212058567825285331, −6.00577018783881464302998237178, −5.25595811340486795022242539778, −3.91081960225713297686344884620, −2.49843760251784489824887836067, 0.53940420580014888750229880594, 2.51597124333535900350697371604, 3.69750109100866306139654995947, 4.40643317764784828151483717622, 6.02188743361029416681630469805, 6.97835036481302262949494162808, 8.422541690071482748630097637856, 9.863286469326729173922861842420, 10.53802059875041053604302511063, 11.38740114303800024670361214879

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