Properties

Label 2-4368-1.1-c1-0-11
Degree $2$
Conductor $4368$
Sign $1$
Analytic cond. $34.8786$
Root an. cond. $5.90581$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2.37·5-s + 7-s + 9-s + 4.37·11-s − 13-s + 2.37·15-s + 4.37·17-s + 2.37·19-s − 21-s − 6.37·23-s + 0.627·25-s − 27-s + 4.37·29-s − 4·31-s − 4.37·33-s − 2.37·35-s + 3.62·37-s + 39-s + 8·41-s + 1.62·43-s − 2.37·45-s − 6.74·47-s + 49-s − 4.37·51-s + 6·53-s − 10.3·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.06·5-s + 0.377·7-s + 0.333·9-s + 1.31·11-s − 0.277·13-s + 0.612·15-s + 1.06·17-s + 0.544·19-s − 0.218·21-s − 1.32·23-s + 0.125·25-s − 0.192·27-s + 0.811·29-s − 0.718·31-s − 0.761·33-s − 0.400·35-s + 0.596·37-s + 0.160·39-s + 1.24·41-s + 0.248·43-s − 0.353·45-s − 0.983·47-s + 0.142·49-s − 0.612·51-s + 0.824·53-s − 1.39·55-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4368,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.344599193\)
\(L(\frac12)\) \(\approx\) \(1.344599193\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.37T + 5T^{2} \)
11 \( 1 - 4.37T + 11T^{2} \)
17 \( 1 - 4.37T + 17T^{2} \)
19 \( 1 - 2.37T + 19T^{2} \)
23 \( 1 + 6.37T + 23T^{2} \)
29 \( 1 - 4.37T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 3.62T + 37T^{2} \)
41 \( 1 - 8T + 41T^{2} \)
43 \( 1 - 1.62T + 43T^{2} \)
47 \( 1 + 6.74T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 6.74T + 59T^{2} \)
61 \( 1 + 5.11T + 61T^{2} \)
67 \( 1 - 8.74T + 67T^{2} \)
71 \( 1 + 6.74T + 71T^{2} \)
73 \( 1 - 0.372T + 73T^{2} \)
79 \( 1 + 4.74T + 79T^{2} \)
83 \( 1 - 2T + 83T^{2} \)
89 \( 1 + 8.74T + 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−8.124601962254785369902042214984, −7.69665069133783332428996171488, −6.98752792048629997736321482333, −6.13419546947065341026689862272, −5.47994746802143115910409983209, −4.42613796915471136825504066282, −4.00899669940403824413288000440, −3.12088017152515090058115103166, −1.70858833268963562478390790873, −0.70137363351857694340745819859, 0.70137363351857694340745819859, 1.70858833268963562478390790873, 3.12088017152515090058115103166, 4.00899669940403824413288000440, 4.42613796915471136825504066282, 5.47994746802143115910409983209, 6.13419546947065341026689862272, 6.98752792048629997736321482333, 7.69665069133783332428996171488, 8.124601962254785369902042214984

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