Properties

Label 2-2475-5.4-c1-0-62
Degree $2$
Conductor $2475$
Sign $-0.447 - 0.894i$
Analytic cond. $19.7629$
Root an. cond. $4.44555$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.41i·2-s − 3.82·4-s − 0.828i·7-s + 4.41i·8-s + 11-s − 5.65i·13-s − 1.99·14-s + 2.99·16-s − 1.17i·17-s + 6.82·19-s − 2.41i·22-s + 4i·23-s − 13.6·26-s + 3.17i·28-s − 4.82·29-s + ⋯
L(s)  = 1  − 1.70i·2-s − 1.91·4-s − 0.313i·7-s + 1.56i·8-s + 0.301·11-s − 1.56i·13-s − 0.534·14-s + 0.749·16-s − 0.284i·17-s + 1.56·19-s − 0.514i·22-s + 0.834i·23-s − 2.67·26-s + 0.599i·28-s − 0.896·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(19.7629\)
Root analytic conductor: \(4.44555\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2475,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.142545298\)
\(L(\frac12)\) \(\approx\) \(1.142545298\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 - T \)
good2 \( 1 + 2.41iT - 2T^{2} \)
7 \( 1 + 0.828iT - 7T^{2} \)
13 \( 1 + 5.65iT - 13T^{2} \)
17 \( 1 + 1.17iT - 17T^{2} \)
19 \( 1 - 6.82T + 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 + 4.82T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 11.6iT - 37T^{2} \)
41 \( 1 + 4.82T + 41T^{2} \)
43 \( 1 + 8.82iT - 43T^{2} \)
47 \( 1 + 4iT - 47T^{2} \)
53 \( 1 + 9.31iT - 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 11.6T + 61T^{2} \)
67 \( 1 - 5.65iT - 67T^{2} \)
71 \( 1 + 2.34T + 71T^{2} \)
73 \( 1 - 11.3iT - 73T^{2} \)
79 \( 1 + 8.48T + 79T^{2} \)
83 \( 1 - 10iT - 83T^{2} \)
89 \( 1 - 3.65T + 89T^{2} \)
97 \( 1 + 11.6iT - 97T^{2} \)
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

−8.786629619200477560483591305558, −7.73626729568590850113583674432, −7.13384223044731206270244568568, −5.59249373788849695434543582942, −5.19714322992353404529127091013, −3.88378118846950798331584440067, −3.44076808623735226929035088474, −2.53329294135711272127591792732, −1.39794245395925405540891681939, −0.41187736469541232322826991742, 1.52319634873200051154830910074, 3.08515366300691333099474903900, 4.33552617551814117429856230038, 4.81682428111886250463378931661, 5.84624012248208243183898944078, 6.38783582181425935419151937416, 7.09601869756324309736631788315, 7.75809995609472801727269168882, 8.531240571515750983246546513408, 9.289395883510053614591950561743

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