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Properties

Label 2-15e2-15.14-c8-0-11
Degree $2$
Conductor $225$
Sign $-0.988 - 0.151i$
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  + 7.31·2-s − 202.·4-s + 3.66e3i·7-s − 3.35e3·8-s + 1.32e4i·11-s + 2.16e4i·13-s + 2.67e4i·14-s + 2.73e4·16-s + 1.17e5·17-s + 2.55e5·19-s + 9.69e4i·22-s − 4.21e5·23-s + 1.58e5i·26-s − 7.41e5i·28-s + 6.10e5i·29-s + ⋯
L(s)  = 1  + 0.457·2-s − 0.791·4-s + 1.52i·7-s − 0.818·8-s + 0.905i·11-s + 0.756i·13-s + 0.697i·14-s + 0.416·16-s + 1.40·17-s + 1.96·19-s + 0.414i·22-s − 1.50·23-s + 0.345i·26-s − 1.20i·28-s + 0.863i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.151i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.988 - 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.988 - 0.151i$
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: \((2,\ 225,\ (\ :4),\ -0.988 - 0.151i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.428800716\)
\(L(\frac12)\) \(\approx\) \(1.428800716\)
\(L(5)\) 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 \( 1 - 7.31T + 256T^{2} \)
7 \( 1 - 3.66e3iT - 5.76e6T^{2} \)
11 \( 1 - 1.32e4iT - 2.14e8T^{2} \)
13 \( 1 - 2.16e4iT - 8.15e8T^{2} \)
17 \( 1 - 1.17e5T + 6.97e9T^{2} \)
19 \( 1 - 2.55e5T + 1.69e10T^{2} \)
23 \( 1 + 4.21e5T + 7.83e10T^{2} \)
29 \( 1 - 6.10e5iT - 5.00e11T^{2} \)
31 \( 1 + 9.80e5T + 8.52e11T^{2} \)
37 \( 1 + 3.00e5iT - 3.51e12T^{2} \)
41 \( 1 - 1.06e6iT - 7.98e12T^{2} \)
43 \( 1 + 1.60e6iT - 1.16e13T^{2} \)
47 \( 1 - 7.57e5T + 2.38e13T^{2} \)
53 \( 1 + 3.30e6T + 6.22e13T^{2} \)
59 \( 1 - 2.09e7iT - 1.46e14T^{2} \)
61 \( 1 - 2.16e7T + 1.91e14T^{2} \)
67 \( 1 - 1.46e7iT - 4.06e14T^{2} \)
71 \( 1 - 1.44e7iT - 6.45e14T^{2} \)
73 \( 1 - 2.16e7iT - 8.06e14T^{2} \)
79 \( 1 + 4.05e7T + 1.51e15T^{2} \)
83 \( 1 + 4.84e7T + 2.25e15T^{2} \)
89 \( 1 + 3.57e7iT - 3.93e15T^{2} \)
97 \( 1 - 6.01e7iT - 7.83e15T^{2} \)
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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

−11.82354484069945080574544942607, −9.919963717276236851681676348360, −9.410855558208125217613516750665, −8.455584950994595049069710212002, −7.29620305206636819416703642424, −5.70439155583080047778119151058, −5.30433506887833848599485738107, −3.97404245491573360265474496442, −2.82828152503101163153827046048, −1.46632455072861663683490364402, 0.33020178364499085892881930066, 1.06647366294584695187860965161, 3.30621471192780903440488216010, 3.79717638105333500971476313642, 5.14725167904267600865281119854, 5.96255259824176511691571369966, 7.54023449828493315889268363455, 8.121085402581688286935877881615, 9.630031277824899796833738220115, 10.15459382263148125214854292412

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