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Properties

Label 2-15e2-15.2-c3-0-16
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: $\chi_{225} (107, \cdot )$
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

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

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