Properties

Label 2-4368-1.1-c1-0-25
Degree $2$
Conductor $4368$
Sign $1$
Analytic cond. $34.8786$
Root an. cond. $5.90581$
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  + 3-s + 7-s + 9-s + 2·11-s − 13-s + 4·17-s − 4·19-s + 21-s − 2·23-s − 5·25-s + 27-s + 6·29-s + 2·33-s + 10·37-s − 39-s + 4·41-s + 4·43-s + 49-s + 4·51-s + 10·53-s − 4·57-s − 4·59-s − 2·61-s + 63-s − 2·69-s + 6·71-s + 14·73-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.603·11-s − 0.277·13-s + 0.970·17-s − 0.917·19-s + 0.218·21-s − 0.417·23-s − 25-s + 0.192·27-s + 1.11·29-s + 0.348·33-s + 1.64·37-s − 0.160·39-s + 0.624·41-s + 0.609·43-s + 1/7·49-s + 0.560·51-s + 1.37·53-s − 0.529·57-s − 0.520·59-s − 0.256·61-s + 0.125·63-s − 0.240·69-s + 0.712·71-s + 1.63·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4368\)    =    \(2^{4} \cdot 3 \cdot 7 \cdot 13\)
Sign: $1$
Analytic conductor: \(34.8786\)
Root analytic conductor: \(5.90581\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4368,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.687163204\)
\(L(\frac12)\) \(\approx\) \(2.687163204\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
7 \( 1 - T \)
13 \( 1 + T \)
good5 \( 1 + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
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

−8.098225820552997776823221528461, −7.958731194267174523471317397116, −6.96586725704390539804768312340, −6.20990946014147515150327787312, −5.45286722771906805968576134016, −4.38001540961947469936916075666, −3.93242944981291926283668604940, −2.83356599595692265996403299665, −2.03533904119451063971207377291, −0.929175110490288964280986467431, 0.929175110490288964280986467431, 2.03533904119451063971207377291, 2.83356599595692265996403299665, 3.93242944981291926283668604940, 4.38001540961947469936916075666, 5.45286722771906805968576134016, 6.20990946014147515150327787312, 6.96586725704390539804768312340, 7.958731194267174523471317397116, 8.098225820552997776823221528461

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