Properties

Label 2-4275-1.1-c1-0-44
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s + 2·7-s − 3·8-s + 4·11-s − 2·13-s + 2·14-s − 16-s − 4·17-s + 19-s + 4·22-s + 6·23-s − 2·26-s − 2·28-s + 6·29-s − 4·31-s + 5·32-s − 4·34-s − 10·37-s + 38-s + 10·41-s + 2·43-s − 4·44-s + 6·46-s + 6·47-s − 3·49-s + 2·52-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 0.755·7-s − 1.06·8-s + 1.20·11-s − 0.554·13-s + 0.534·14-s − 1/4·16-s − 0.970·17-s + 0.229·19-s + 0.852·22-s + 1.25·23-s − 0.392·26-s − 0.377·28-s + 1.11·29-s − 0.718·31-s + 0.883·32-s − 0.685·34-s − 1.64·37-s + 0.162·38-s + 1.56·41-s + 0.304·43-s − 0.603·44-s + 0.884·46-s + 0.875·47-s − 3/7·49-s + 0.277·52-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4275,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.477243724\)
\(L(\frac12)\) \(\approx\) \(2.477243724\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 18 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.561770733228379757186849350649, −7.59558320917550972465555490812, −6.78501474904600429867238246747, −6.14092725738521158012724144633, −5.11486498158351745998026612688, −4.73128048489984459968975866113, −3.96443725324887016941416908170, −3.14829915765960191109349093245, −2.05228468867020290309731494906, −0.821382376766833850700942871667, 0.821382376766833850700942871667, 2.05228468867020290309731494906, 3.14829915765960191109349093245, 3.96443725324887016941416908170, 4.73128048489984459968975866113, 5.11486498158351745998026612688, 6.14092725738521158012724144633, 6.78501474904600429867238246747, 7.59558320917550972465555490812, 8.561770733228379757186849350649

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