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Properties

Label 2-15e2-15.14-c8-0-13
Degree $2$
Conductor $225$
Sign $0.472 - 0.881i$
Analytic cond. $91.6601$
Root an. cond. $9.57393$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 13.3·2-s − 76.9·4-s − 256. i·7-s − 4.45e3·8-s − 2.89e4i·11-s + 4.59e4i·13-s − 3.42e3i·14-s − 3.99e4·16-s + 2.72e4·17-s − 1.26e5·19-s − 3.87e5i·22-s − 1.41e5·23-s + 6.14e5i·26-s + 1.97e4i·28-s − 6.58e5i·29-s + ⋯
L(s)  = 1  + 0.836·2-s − 0.300·4-s − 0.106i·7-s − 1.08·8-s − 1.97i·11-s + 1.60i·13-s − 0.0892i·14-s − 0.609·16-s + 0.326·17-s − 0.972·19-s − 1.65i·22-s − 0.504·23-s + 1.34i·26-s + 0.0320i·28-s − 0.931i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.472 - 0.881i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.472 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.472 - 0.881i$
Analytic conductor: \(91.6601\)
Root analytic conductor: \(9.57393\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (224, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :4),\ 0.472 - 0.881i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.816682435\)
\(L(\frac12)\) \(\approx\) \(1.816682435\)
\(L(5)\) 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 \( 1 - 13.3T + 256T^{2} \)
7 \( 1 + 256. iT - 5.76e6T^{2} \)
11 \( 1 + 2.89e4iT - 2.14e8T^{2} \)
13 \( 1 - 4.59e4iT - 8.15e8T^{2} \)
17 \( 1 - 2.72e4T + 6.97e9T^{2} \)
19 \( 1 + 1.26e5T + 1.69e10T^{2} \)
23 \( 1 + 1.41e5T + 7.83e10T^{2} \)
29 \( 1 + 6.58e5iT - 5.00e11T^{2} \)
31 \( 1 - 1.39e6T + 8.52e11T^{2} \)
37 \( 1 - 5.16e5iT - 3.51e12T^{2} \)
41 \( 1 - 9.16e3iT - 7.98e12T^{2} \)
43 \( 1 - 5.24e6iT - 1.16e13T^{2} \)
47 \( 1 + 5.56e6T + 2.38e13T^{2} \)
53 \( 1 - 4.27e6T + 6.22e13T^{2} \)
59 \( 1 - 1.18e7iT - 1.46e14T^{2} \)
61 \( 1 + 6.05e6T + 1.91e14T^{2} \)
67 \( 1 - 1.97e7iT - 4.06e14T^{2} \)
71 \( 1 - 8.03e6iT - 6.45e14T^{2} \)
73 \( 1 - 4.28e7iT - 8.06e14T^{2} \)
79 \( 1 - 4.31e7T + 1.51e15T^{2} \)
83 \( 1 + 9.09e6T + 2.25e15T^{2} \)
89 \( 1 - 7.95e7iT - 3.93e15T^{2} \)
97 \( 1 + 5.87e7iT - 7.83e15T^{2} \)
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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

−11.33057559032595432181419039336, −9.978289628053079416129827793122, −8.877949517071589520621424004404, −8.212440708063275046442492127084, −6.47845728698227441015498362971, −5.90363681965108246237513480878, −4.56590560511709653512180981264, −3.77959600658669025737982995548, −2.61594642055286928406662080681, −0.889479258979031095416581187923, 0.37991887864988229348008826473, 2.09419730053160027359646605556, 3.32146943238980202213670368024, 4.49012314892241842618788291098, 5.22706762555514874300112666090, 6.35849648692891804840250263733, 7.58504469019197432863384130801, 8.641794141325892666686302267063, 9.820758026395228847441217796924, 10.49095914170417395595817185772

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