↑

Properties

Label 2-15e2-9.7-c3-0-34
Degree $2$
Conductor $225$
Sign $0.982 + 0.185i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.392 − 0.680i)2-s + (3.69 − 3.65i)3-s + (3.69 + 6.39i)4-s + (−1.03 − 3.94i)6-s + (−10.4 + 18.1i)7-s + 12.0·8-s + (0.319 − 26.9i)9-s + (29.6 − 51.4i)11-s + (36.9 + 10.1i)12-s + (22.9 + 39.7i)13-s + (8.21 + 14.2i)14-s + (−24.7 + 42.9i)16-s + 43.0·17-s + (−18.2 − 10.8i)18-s + 140.·19-s + ⋯
L(s)  = 1  + (0.138 − 0.240i)2-s + (0.711 − 0.702i)3-s + (0.461 + 0.799i)4-s + (−0.0702 − 0.268i)6-s + (−0.564 + 0.977i)7-s + 0.533·8-s + (0.0118 − 0.999i)9-s + (0.813 − 1.40i)11-s + (0.890 + 0.244i)12-s + (0.489 + 0.848i)13-s + (0.156 + 0.271i)14-s + (−0.387 + 0.670i)16-s + 0.614·17-s + (−0.238 − 0.141i)18-s + 1.69·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.185i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.982 + 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.982 + 0.185i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 0.982 + 0.185i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.80570 - 0.262211i\)
\(L(\frac12)\) \(\approx\) \(2.80570 - 0.262211i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.69 + 3.65i)T \)
5 \( 1 \)
good2 \( 1 + (-0.392 + 0.680i)T + (-4 - 6.92i)T^{2} \)
7 \( 1 + (10.4 - 18.1i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-29.6 + 51.4i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-22.9 - 39.7i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 43.0T + 4.91e3T^{2} \)
19 \( 1 - 140.T + 6.85e3T^{2} \)
23 \( 1 + (-50.8 - 88.1i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (6.19 - 10.7i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (35.9 + 62.3i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 150.T + 5.06e4T^{2} \)
41 \( 1 + (25.5 + 44.2i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-17.2 + 29.8i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (58.9 - 102. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 137.T + 1.48e5T^{2} \)
59 \( 1 + (248. + 429. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-123. + 214. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (413. + 715. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 260.T + 3.57e5T^{2} \)
73 \( 1 + 372.T + 3.89e5T^{2} \)
79 \( 1 + (238. - 412. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (156. - 271. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 817.T + 7.04e5T^{2} \)
97 \( 1 + (470. - 815. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.82345758778394747504976371882, −11.32221202203697153789413098248, −9.430211521498548821343134186021, −8.825800663762802023416914724534, −7.81551997948105240489132548584, −6.76949852389841408416086917339, −5.81193391430525962094952800964, −3.57922752009375907075071740062, −3.02675180812574767124405873709, −1.46975240088237198179019257562, 1.31023123635619362921773029288, 3.10252341703793245357876189026, 4.34527104662344193364645788059, 5.46583688059554030707177314581, 6.92558749116553037632627435678, 7.59091381756785575504959353342, 9.144823186629140705212060628666, 10.09912872383895789119406180621, 10.37624058999068875682861267715, 11.69641580683652246712702486440

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