↑

Properties

Label 2-15e2-5.4-c7-0-17
Degree $2$
Conductor $225$
Sign $-0.447 - 0.894i$
Analytic cond. $70.2866$
Root an. cond. $8.38371$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 6i·2-s + 92·4-s + 64i·7-s + 1.32e3i·8-s + 948·11-s − 5.09e3i·13-s − 384·14-s + 3.85e3·16-s + 2.83e4i·17-s + 8.62e3·19-s + 5.68e3i·22-s + 1.52e4i·23-s + 3.05e4·26-s + 5.88e3i·28-s + 3.65e4·29-s + ⋯
L(s)  = 1  + 0.530i·2-s + 0.718·4-s + 0.0705i·7-s + 0.911i·8-s + 0.214·11-s − 0.643i·13-s − 0.0374·14-s + 0.235·16-s + 1.40i·17-s + 0.288·19-s + 0.113i·22-s + 0.262i·23-s + 0.341·26-s + 0.0506i·28-s + 0.277·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(70.2866\)
Root analytic conductor: \(8.38371\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :7/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.380285569\)
\(L(\frac12)\) \(\approx\) \(2.380285569\)
\(L(\frac{9}{2})\) 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 \( 1 - 6iT - 128T^{2} \)
7 \( 1 - 64iT - 8.23e5T^{2} \)
11 \( 1 - 948T + 1.94e7T^{2} \)
13 \( 1 + 5.09e3iT - 6.27e7T^{2} \)
17 \( 1 - 2.83e4iT - 4.10e8T^{2} \)
19 \( 1 - 8.62e3T + 8.93e8T^{2} \)
23 \( 1 - 1.52e4iT - 3.40e9T^{2} \)
29 \( 1 - 3.65e4T + 1.72e10T^{2} \)
31 \( 1 + 2.76e5T + 2.75e10T^{2} \)
37 \( 1 + 2.68e5iT - 9.49e10T^{2} \)
41 \( 1 - 6.29e5T + 1.94e11T^{2} \)
43 \( 1 - 6.85e5iT - 2.71e11T^{2} \)
47 \( 1 - 5.83e5iT - 5.06e11T^{2} \)
53 \( 1 - 4.28e5iT - 1.17e12T^{2} \)
59 \( 1 - 1.30e6T + 2.48e12T^{2} \)
61 \( 1 - 3.00e5T + 3.14e12T^{2} \)
67 \( 1 - 5.07e5iT - 6.06e12T^{2} \)
71 \( 1 + 5.56e6T + 9.09e12T^{2} \)
73 \( 1 - 1.36e6iT - 1.10e13T^{2} \)
79 \( 1 - 6.91e6T + 1.92e13T^{2} \)
83 \( 1 - 4.37e6iT - 2.71e13T^{2} \)
89 \( 1 + 8.52e6T + 4.42e13T^{2} \)
97 \( 1 - 8.82e6iT - 8.07e13T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.15105095658624620173158475341, −10.52694459073778741584251315878, −9.217794542987190426455789259240, −8.091605807919504353675571647858, −7.33720912230895446627296708931, −6.17850407807383035821864066716, −5.46499258685382484333402600151, −3.87555893189226680981403347497, −2.55003598672261411497073527977, −1.29531014608654992542475489573, 0.55438036008380261228286944237, 1.82632877636914417024830143450, 2.90505598717356100377961163292, 4.08716755828196382330466598274, 5.50126743160538765753684676405, 6.79156762869029948438089217545, 7.43186669446770798760963195365, 8.929103900416934367891360769240, 9.814465001916583447003026420218, 10.82280912247434167234650656206

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