Properties

Label 2-2475-5.4-c1-0-65
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  − 0.414i·2-s + 1.82·4-s − 4.82i·7-s − 1.58i·8-s + 11-s − 5.65i·13-s − 1.99·14-s + 3·16-s + 6.82i·17-s + 1.17·19-s − 0.414i·22-s − 4i·23-s − 2.34·26-s − 8.82i·28-s + 0.828·29-s + ⋯
L(s)  = 1  − 0.292i·2-s + 0.914·4-s − 1.82i·7-s − 0.560i·8-s + 0.301·11-s − 1.56i·13-s − 0.534·14-s + 0.750·16-s + 1.65i·17-s + 0.268·19-s − 0.0883i·22-s − 0.834i·23-s − 0.459·26-s − 1.66i·28-s + 0.153·29-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

\(L(1)\) \(\approx\) \(2.227335406\)
\(L(\frac12)\) \(\approx\) \(2.227335406\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.414iT - 2T^{2} \)
7 \( 1 + 4.82iT - 7T^{2} \)
13 \( 1 + 5.65iT - 13T^{2} \)
17 \( 1 - 6.82iT - 17T^{2} \)
19 \( 1 - 1.17T + 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 - 0.828T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 0.343iT - 37T^{2} \)
41 \( 1 - 0.828T + 41T^{2} \)
43 \( 1 - 3.17iT - 43T^{2} \)
47 \( 1 - 4iT - 47T^{2} \)
53 \( 1 + 13.3iT - 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 0.343T + 61T^{2} \)
67 \( 1 - 5.65iT - 67T^{2} \)
71 \( 1 + 13.6T + 71T^{2} \)
73 \( 1 - 11.3iT - 73T^{2} \)
79 \( 1 - 8.48T + 79T^{2} \)
83 \( 1 + 10iT - 83T^{2} \)
89 \( 1 + 7.65T + 89T^{2} \)
97 \( 1 - 0.343iT - 97T^{2} \)
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

−8.412356141346502090162147934824, −7.86756777053947113082711152536, −7.16583634226212540534802536721, −6.46342116206719107276429639516, −5.73112847783807713090698179165, −4.46900875007296364615234843844, −3.68486096630303576990780475333, −2.99461009951280061993440136305, −1.62503927732425910002157252449, −0.72191048936972842072221422669, 1.65623609262533898635247304361, 2.44135873925448570708139591146, 3.20794363665649943270852398275, 4.62629272554037818486928924440, 5.45866781020379517862430150697, 6.08661421270061694517609354089, 6.86741632614896323544157588467, 7.48523824736595798064823709875, 8.470684269732932870276020766259, 9.293320117457178538631982482247

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