Properties

Label 2-4275-1.1-c1-0-114
Degree $2$
Conductor $4275$
Sign $-1$
Analytic cond. $34.1360$
Root an. cond. $5.84260$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s + 2·7-s + 3·8-s + 6·11-s − 2·14-s − 16-s − 6·17-s + 19-s − 6·22-s − 8·23-s − 2·28-s − 4·29-s − 5·32-s + 6·34-s − 4·37-s − 38-s + 2·43-s − 6·44-s + 8·46-s − 8·47-s − 3·49-s + 2·53-s + 6·56-s + 4·58-s − 12·59-s + 2·61-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.755·7-s + 1.06·8-s + 1.80·11-s − 0.534·14-s − 1/4·16-s − 1.45·17-s + 0.229·19-s − 1.27·22-s − 1.66·23-s − 0.377·28-s − 0.742·29-s − 0.883·32-s + 1.02·34-s − 0.657·37-s − 0.162·38-s + 0.304·43-s − 0.904·44-s + 1.17·46-s − 1.16·47-s − 3/7·49-s + 0.274·53-s + 0.801·56-s + 0.525·58-s − 1.56·59-s + 0.256·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4275\)    =    \(3^{2} \cdot 5^{2} \cdot 19\)
Sign: $-1$
Analytic conductor: \(34.1360\)
Root analytic conductor: \(5.84260\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{4275} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4275,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19 \( 1 - T \)
good2 \( 1 + T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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

−8.154904276278665956764754494175, −7.49006725304513024040398275037, −6.65598056508002457182870521609, −5.94645344090271988207277046134, −4.82592459399091433680554684550, −4.27331149587142879061517048995, −3.60839819994467751886447584009, −1.98429554072259245310201630187, −1.40340994003836782410675250116, 0, 1.40340994003836782410675250116, 1.98429554072259245310201630187, 3.60839819994467751886447584009, 4.27331149587142879061517048995, 4.82592459399091433680554684550, 5.94645344090271988207277046134, 6.65598056508002457182870521609, 7.49006725304513024040398275037, 8.154904276278665956764754494175

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