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Properties

Label 2-15e2-5.4-c3-0-19
Degree $2$
Conductor $225$
Sign $-0.894 - 0.447i$
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  − 3.16i·2-s − 2.00·4-s + 15i·7-s − 18.9i·8-s − 63.2·11-s − 35i·13-s + 47.4·14-s − 76·16-s − 88.5i·17-s − 91·19-s + 200i·22-s − 113. i·23-s − 110.·26-s − 30.0i·28-s − 63.2·29-s + ⋯
L(s)  = 1  − 1.11i·2-s − 0.250·4-s + 0.809i·7-s − 0.838i·8-s − 1.73·11-s − 0.746i·13-s + 0.905·14-s − 1.18·16-s − 1.26i·17-s − 1.09·19-s + 1.93i·22-s − 1.03i·23-s − 0.834·26-s − 0.202i·28-s − 0.404·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.187073 + 0.792454i\)
\(L(\frac12)\) \(\approx\) \(0.187073 + 0.792454i\)
\(L(\frac{5}{2})\) 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 \( 1 + 3.16iT - 8T^{2} \)
7 \( 1 - 15iT - 343T^{2} \)
11 \( 1 + 63.2T + 1.33e3T^{2} \)
13 \( 1 + 35iT - 2.19e3T^{2} \)
17 \( 1 + 88.5iT - 4.91e3T^{2} \)
19 \( 1 + 91T + 6.85e3T^{2} \)
23 \( 1 + 113. iT - 1.21e4T^{2} \)
29 \( 1 + 63.2T + 2.43e4T^{2} \)
31 \( 1 + 147T + 2.97e4T^{2} \)
37 \( 1 - 370iT - 5.06e4T^{2} \)
41 \( 1 - 442.T + 6.89e4T^{2} \)
43 \( 1 + 335iT - 7.95e4T^{2} \)
47 \( 1 - 177. iT - 1.03e5T^{2} \)
53 \( 1 - 88.5iT - 1.48e5T^{2} \)
59 \( 1 + 885.T + 2.05e5T^{2} \)
61 \( 1 - 427T + 2.26e5T^{2} \)
67 \( 1 - 15iT - 3.00e5T^{2} \)
71 \( 1 - 63.2T + 3.57e5T^{2} \)
73 \( 1 - 70iT - 3.89e5T^{2} \)
79 \( 1 - 876T + 4.93e5T^{2} \)
83 \( 1 + 531. iT - 5.71e5T^{2} \)
89 \( 1 + 7.04e5T^{2} \)
97 \( 1 + 1.08e3iT - 9.12e5T^{2} \)
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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

−11.08878363009961207328711715637, −10.54635185491323010235097135491, −9.571936976051561855465745118854, −8.443434834850680752285119116856, −7.29784914026726195275685132207, −5.86598827803931735026584334301, −4.69474076164117674731632221597, −2.97943474939384084206824540315, −2.29879582618092112555991173880, −0.30324830918853772506236517044, 2.15565980484628573858913676225, 4.04624084963403353181420888668, 5.34509012550201650055277875612, 6.29775069127369547833828267176, 7.46135928998467572538510548669, 7.960969285221124169398637566560, 9.177668458964042154196522476501, 10.62514850568218668104245782257, 11.04711005982576152721063725798, 12.64616772570852353134770951958

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