Properties

Label 2-275-5.4-c5-0-30
Degree $2$
Conductor $275$
Sign $0.447 - 0.894i$
Analytic cond. $44.1055$
Root an. cond. $6.64120$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.64i·2-s − 6.66i·3-s + 25.0·4-s + 17.6·6-s − 12.8i·7-s + 150. i·8-s + 198.·9-s − 121·11-s − 166. i·12-s + 485. i·13-s + 34.0·14-s + 402.·16-s + 266. i·17-s + 524. i·18-s + 149.·19-s + ⋯
L(s)  = 1  + 0.467i·2-s − 0.427i·3-s + 0.781·4-s + 0.199·6-s − 0.0993i·7-s + 0.832i·8-s + 0.817·9-s − 0.301·11-s − 0.334i·12-s + 0.796i·13-s + 0.0464·14-s + 0.393·16-s + 0.223i·17-s + 0.381i·18-s + 0.0951·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(44.1055\)
Root analytic conductor: \(6.64120\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :5/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.675516728\)
\(L(\frac12)\) \(\approx\) \(2.675516728\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 + 121T \)
good2 \( 1 - 2.64iT - 32T^{2} \)
3 \( 1 + 6.66iT - 243T^{2} \)
7 \( 1 + 12.8iT - 1.68e4T^{2} \)
13 \( 1 - 485. iT - 3.71e5T^{2} \)
17 \( 1 - 266. iT - 1.41e6T^{2} \)
19 \( 1 - 149.T + 2.47e6T^{2} \)
23 \( 1 - 3.21e3iT - 6.43e6T^{2} \)
29 \( 1 + 2.94e3T + 2.05e7T^{2} \)
31 \( 1 - 2.14e3T + 2.86e7T^{2} \)
37 \( 1 + 808. iT - 6.93e7T^{2} \)
41 \( 1 - 1.01e4T + 1.15e8T^{2} \)
43 \( 1 + 2.76e3iT - 1.47e8T^{2} \)
47 \( 1 - 9.97e3iT - 2.29e8T^{2} \)
53 \( 1 + 7.12e3iT - 4.18e8T^{2} \)
59 \( 1 - 3.33e4T + 7.14e8T^{2} \)
61 \( 1 + 1.18e4T + 8.44e8T^{2} \)
67 \( 1 - 4.50e3iT - 1.35e9T^{2} \)
71 \( 1 + 4.59e4T + 1.80e9T^{2} \)
73 \( 1 - 6.20e4iT - 2.07e9T^{2} \)
79 \( 1 - 5.74e4T + 3.07e9T^{2} \)
83 \( 1 - 9.05e4iT - 3.93e9T^{2} \)
89 \( 1 - 1.27e5T + 5.58e9T^{2} \)
97 \( 1 - 1.32e5iT - 8.58e9T^{2} \)
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

−11.30279682461492255176931714030, −10.32650941680355138577753020416, −9.253963619401727689833901485735, −7.922438866456434540363608900545, −7.27027121300514872909793266185, −6.45015087866668184470010792294, −5.37760830299972027698839881884, −3.93986942825683408782698337178, −2.36109517289819793345399183756, −1.29683034369914266890006526858, 0.76839829591370221553046777192, 2.19388447549496415532536776974, 3.31864610098420806238837035564, 4.52901272967868245011212643993, 5.83780649473672942767158009495, 6.97720110144316707152635017451, 7.86769904453670043029204676807, 9.208216449573200848616407382887, 10.29226562029504091669297820816, 10.63298795128666217242627103550

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