Properties

Label 2-15e2-225.4-c3-0-83
Degree $2$
Conductor $225$
Sign $0.00670 - 0.999i$
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  + (−1.01 − 4.78i)2-s + (3.24 + 4.05i)3-s + (−14.5 + 6.45i)4-s + (−5.06 − 9.96i)5-s + (16.1 − 19.6i)6-s + (11.0 − 6.37i)7-s + (22.6 + 31.1i)8-s + (−5.93 + 26.3i)9-s + (−42.4 + 34.3i)10-s + (−32.9 + 6.99i)11-s + (−73.2 − 37.9i)12-s + (−0.362 + 1.70i)13-s + (−41.6 − 46.2i)14-s + (23.9 − 52.9i)15-s + (40.9 − 45.4i)16-s + (−75.1 − 103. i)17-s + ⋯
L(s)  = 1  + (−0.359 − 1.69i)2-s + (0.624 + 0.780i)3-s + (−1.81 + 0.807i)4-s + (−0.453 − 0.891i)5-s + (1.09 − 1.33i)6-s + (0.595 − 0.344i)7-s + (1.00 + 1.37i)8-s + (−0.219 + 0.975i)9-s + (−1.34 + 1.08i)10-s + (−0.902 + 0.191i)11-s + (−1.76 − 0.912i)12-s + (−0.00772 + 0.0363i)13-s + (−0.795 − 0.883i)14-s + (0.412 − 0.910i)15-s + (0.639 − 0.710i)16-s + (−1.07 − 1.47i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00670 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.00670 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.00670 - 0.999i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 0.00670 - 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0865817 + 0.0860033i\)
\(L(\frac12)\) \(\approx\) \(0.0865817 + 0.0860033i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.24 - 4.05i)T \)
5 \( 1 + (5.06 + 9.96i)T \)
good2 \( 1 + (1.01 + 4.78i)T + (-7.30 + 3.25i)T^{2} \)
7 \( 1 + (-11.0 + 6.37i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (32.9 - 6.99i)T + (1.21e3 - 541. i)T^{2} \)
13 \( 1 + (0.362 - 1.70i)T + (-2.00e3 - 893. i)T^{2} \)
17 \( 1 + (75.1 + 103. i)T + (-1.51e3 + 4.67e3i)T^{2} \)
19 \( 1 + (93.2 - 67.7i)T + (2.11e3 - 6.52e3i)T^{2} \)
23 \( 1 + (-42.4 + 38.2i)T + (1.27e3 - 1.21e4i)T^{2} \)
29 \( 1 + (19.2 - 183. i)T + (-2.38e4 - 5.07e3i)T^{2} \)
31 \( 1 + (-6.00 - 57.1i)T + (-2.91e4 + 6.19e3i)T^{2} \)
37 \( 1 + (-2.72 + 0.884i)T + (4.09e4 - 2.97e4i)T^{2} \)
41 \( 1 + (-5.01 - 1.06i)T + (6.29e4 + 2.80e4i)T^{2} \)
43 \( 1 + (177. - 102. i)T + (3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-212. - 22.2i)T + (1.01e5 + 2.15e4i)T^{2} \)
53 \( 1 + (-180. + 248. i)T + (-4.60e4 - 1.41e5i)T^{2} \)
59 \( 1 + (775. + 164. i)T + (1.87e5 + 8.35e4i)T^{2} \)
61 \( 1 + (-429. + 91.2i)T + (2.07e5 - 9.23e4i)T^{2} \)
67 \( 1 + (337. - 35.4i)T + (2.94e5 - 6.25e4i)T^{2} \)
71 \( 1 + (772. + 560. i)T + (1.10e5 + 3.40e5i)T^{2} \)
73 \( 1 + (781. + 254. i)T + (3.14e5 + 2.28e5i)T^{2} \)
79 \( 1 + (-91.6 + 872. i)T + (-4.82e5 - 1.02e5i)T^{2} \)
83 \( 1 + (107. - 240. i)T + (-3.82e5 - 4.24e5i)T^{2} \)
89 \( 1 + (-359. + 1.10e3i)T + (-5.70e5 - 4.14e5i)T^{2} \)
97 \( 1 + (-1.33e3 - 140. i)T + (8.92e5 + 1.89e5i)T^{2} \)
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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

−10.91274717064602489558232013121, −10.31613387219272415055840389070, −9.165371859018923198499941607723, −8.640381151419249207918739769618, −7.65026884445531509225645836369, −4.89234086419489808153118978937, −4.41263959591153951582441953266, −3.09319981025149691175777467038, −1.82338455969408966940876894671, −0.05027605016384276030105979361, 2.39326380905585489331695744017, 4.25882849548837660231178247566, 5.86525004644693108827437641130, 6.65949105637355484295752965734, 7.56527003309583218072427937454, 8.282533193408802210960008176251, 8.902286540242987153692016540017, 10.41294945631994905369284682945, 11.52750307780690726895300899377, 13.01670208062227901603193305784

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