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Properties

Label 2-15e2-5.2-c2-0-5
Degree $2$
Conductor $225$
Sign $-0.525 + 0.850i$
Analytic cond. $6.13080$
Root an. cond. $2.47604$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.22 − 2.22i)2-s + 5.89i·4-s + (1.44 + 1.44i)7-s + (4.22 − 4.22i)8-s + 3.34·11-s + (10.4 − 10.4i)13-s − 6.44i·14-s + 4.79·16-s + (−2.65 − 2.65i)17-s − 20.6i·19-s + (−7.44 − 7.44i)22-s + (16.4 − 16.4i)23-s − 46.4·26-s + (−8.55 + 8.55i)28-s − 0.853i·29-s + ⋯
L(s)  = 1  + (−1.11 − 1.11i)2-s + 1.47i·4-s + (0.207 + 0.207i)7-s + (0.528 − 0.528i)8-s + 0.304·11-s + (0.803 − 0.803i)13-s − 0.460i·14-s + 0.299·16-s + (−0.155 − 0.155i)17-s − 1.08i·19-s + (−0.338 − 0.338i)22-s + (0.715 − 0.715i)23-s − 1.78·26-s + (−0.305 + 0.305i)28-s − 0.0294i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.525 + 0.850i$
Analytic conductor: \(6.13080\)
Root analytic conductor: \(2.47604\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1),\ -0.525 + 0.850i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.398065 - 0.713972i\)
\(L(\frac12)\) \(\approx\) \(0.398065 - 0.713972i\)
\(L(2)\) 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 \( 1 + (2.22 + 2.22i)T + 4iT^{2} \)
7 \( 1 + (-1.44 - 1.44i)T + 49iT^{2} \)
11 \( 1 - 3.34T + 121T^{2} \)
13 \( 1 + (-10.4 + 10.4i)T - 169iT^{2} \)
17 \( 1 + (2.65 + 2.65i)T + 289iT^{2} \)
19 \( 1 + 20.6iT - 361T^{2} \)
23 \( 1 + (-16.4 + 16.4i)T - 529iT^{2} \)
29 \( 1 + 0.853iT - 841T^{2} \)
31 \( 1 + 18.6T + 961T^{2} \)
37 \( 1 + (38.0 + 38.0i)T + 1.36e3iT^{2} \)
41 \( 1 - 28.6T + 1.68e3T^{2} \)
43 \( 1 + (22.4 - 22.4i)T - 1.84e3iT^{2} \)
47 \( 1 + (-19.7 - 19.7i)T + 2.20e3iT^{2} \)
53 \( 1 + (-28.6 + 28.6i)T - 2.80e3iT^{2} \)
59 \( 1 + 111. iT - 3.48e3T^{2} \)
61 \( 1 - 94.0T + 3.72e3T^{2} \)
67 \( 1 + (-54.8 - 54.8i)T + 4.48e3iT^{2} \)
71 \( 1 - 68T + 5.04e3T^{2} \)
73 \( 1 + (-39.7 + 39.7i)T - 5.32e3iT^{2} \)
79 \( 1 - 24.4iT - 6.24e3T^{2} \)
83 \( 1 + (21.1 - 21.1i)T - 6.88e3iT^{2} \)
89 \( 1 - 94.1iT - 7.92e3T^{2} \)
97 \( 1 + (14.5 + 14.5i)T + 9.40e3iT^{2} \)
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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

−11.24468526786621268215995882263, −10.90203957079898620604456906206, −9.769599786992357123059331924040, −8.878998384723561650629994747399, −8.203311516710162172943877441495, −6.87122651976291343758957813108, −5.32004110061173010171602110949, −3.59342923529175953495772701220, −2.31519511012120369943232164673, −0.73356309336395783174297990099, 1.37564493382491436996024888761, 3.81519427864255003727833129861, 5.49611424865342524572353321773, 6.53357725412856894096610605873, 7.38081347429198476107736836873, 8.424693531100273962988957140831, 9.122299994711475766299016825341, 10.11354558556224987759025186306, 11.10097215576295174579228456672, 12.23112600750048122891386423380

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