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Properties

Label 2-15e2-45.34-c3-0-19
Degree $2$
Conductor $225$
Sign $-0.544 + 0.838i$
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  + (−3.78 − 2.18i)2-s + (−3.78 − 3.55i)3-s + (5.55 + 9.62i)4-s + (6.55 + 21.7i)6-s + (−10.4 − 6.05i)7-s − 13.6i·8-s + (1.67 + 26.9i)9-s + (−5.01 + 8.67i)11-s + (13.2 − 56.2i)12-s + (42.0 − 24.2i)13-s + (26.4 + 45.8i)14-s + (14.6 − 25.4i)16-s + 75.3i·17-s + (52.5 − 105. i)18-s + 116.·19-s + ⋯
L(s)  = 1  + (−1.33 − 0.772i)2-s + (−0.728 − 0.684i)3-s + (0.694 + 1.20i)4-s + (0.446 + 1.48i)6-s + (−0.566 − 0.327i)7-s − 0.602i·8-s + (0.0620 + 0.998i)9-s + (−0.137 + 0.237i)11-s + (0.317 − 1.35i)12-s + (0.897 − 0.518i)13-s + (0.505 + 0.875i)14-s + (0.229 − 0.397i)16-s + 1.07i·17-s + (0.688 − 1.38i)18-s + 1.40·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.544 + 0.838i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.544 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.544 + 0.838i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ -0.544 + 0.838i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.256974 - 0.473364i\)
\(L(\frac12)\) \(\approx\) \(0.256974 - 0.473364i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.78 + 3.55i)T \)
5 \( 1 \)
good2 \( 1 + (3.78 + 2.18i)T + (4 + 6.92i)T^{2} \)
7 \( 1 + (10.4 + 6.05i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (5.01 - 8.67i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-42.0 + 24.2i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 75.3iT - 4.91e3T^{2} \)
19 \( 1 - 116.T + 6.85e3T^{2} \)
23 \( 1 + (-32.9 + 19.0i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (11.3 - 19.5i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (15.0 + 26.0i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 130. iT - 5.06e4T^{2} \)
41 \( 1 + (173. + 300. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-23.1 - 13.3i)T + (3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (399. + 230. i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 438. iT - 1.48e5T^{2} \)
59 \( 1 + (-4.18 - 7.24i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-41.0 + 71.0i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-591. + 341. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 1.09e3T + 3.57e5T^{2} \)
73 \( 1 + 470. iT - 3.89e5T^{2} \)
79 \( 1 + (-243. + 420. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-85.8 - 49.5i)T + (2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 8.80T + 7.04e5T^{2} \)
97 \( 1 + (572. + 330. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−11.23450084022362577859071713963, −10.52482130872114753082626964786, −9.764180753408549930128919727987, −8.512287412012833201781767805562, −7.68590600784173831840656110764, −6.62565546404034724125362186670, −5.36976591181019037232046248222, −3.33153356750053739117598617720, −1.71108871524131683433206996975, −0.55613307329058297484414500778, 0.902841573754117273075248849074, 3.44597991474239801634987145235, 5.18316849279686111481788013256, 6.23727935498628076455611463287, 7.04576186929529294323427938210, 8.328936804600581811259009239123, 9.472266392250015418768374646274, 9.671125807920650662328242039033, 10.98320995250973467514194805404, 11.66334286728574944280355968963

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