Properties

Label 2-4368-1.1-c1-0-36
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 3.81·5-s − 7-s + 9-s + 4.73·11-s + 13-s + 3.81·15-s − 5.22·17-s − 2.92·19-s + 21-s − 3.33·23-s + 9.55·25-s − 27-s − 0.922·29-s + 7.51·31-s − 4.73·33-s + 3.81·35-s + 0.154·37-s − 39-s + 6.36·41-s + 6.55·43-s − 3.81·45-s − 9.03·47-s + 49-s + 5.22·51-s + 8.55·53-s − 18.0·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.70·5-s − 0.377·7-s + 0.333·9-s + 1.42·11-s + 0.277·13-s + 0.984·15-s − 1.26·17-s − 0.670·19-s + 0.218·21-s − 0.694·23-s + 1.91·25-s − 0.192·27-s − 0.171·29-s + 1.35·31-s − 0.824·33-s + 0.644·35-s + 0.0254·37-s − 0.160·39-s + 0.994·41-s + 0.999·43-s − 0.568·45-s − 1.31·47-s + 0.142·49-s + 0.731·51-s + 1.17·53-s − 2.43·55-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: \(1\)
Selberg data: \((2,\ 4368,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 \)
3 \( 1 + T \)
7 \( 1 + T \)
13 \( 1 - T \)
good5 \( 1 + 3.81T + 5T^{2} \)
11 \( 1 - 4.73T + 11T^{2} \)
17 \( 1 + 5.22T + 17T^{2} \)
19 \( 1 + 2.92T + 19T^{2} \)
23 \( 1 + 3.33T + 23T^{2} \)
29 \( 1 + 0.922T + 29T^{2} \)
31 \( 1 - 7.51T + 31T^{2} \)
37 \( 1 - 0.154T + 37T^{2} \)
41 \( 1 - 6.36T + 41T^{2} \)
43 \( 1 - 6.55T + 43T^{2} \)
47 \( 1 + 9.03T + 47T^{2} \)
53 \( 1 - 8.55T + 53T^{2} \)
59 \( 1 + 3.95T + 59T^{2} \)
61 \( 1 - 12.4T + 61T^{2} \)
67 \( 1 - 10.6T + 67T^{2} \)
71 \( 1 - 6.58T + 71T^{2} \)
73 \( 1 + 7.73T + 73T^{2} \)
79 \( 1 + 13.3T + 79T^{2} \)
83 \( 1 - 1.40T + 83T^{2} \)
89 \( 1 + 1.96T + 89T^{2} \)
97 \( 1 + 2.11T + 97T^{2} \)
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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

−8.069911760640105394766209541968, −7.10553704569394816009048973191, −6.64078503298745854384165795732, −6.01828820532158172226743560416, −4.75927787377057189179851769076, −4.05272872141826898148252331157, −3.81646177768785286233772553466, −2.51087080987043957889307459634, −1.06388457822202578151231961597, 0, 1.06388457822202578151231961597, 2.51087080987043957889307459634, 3.81646177768785286233772553466, 4.05272872141826898148252331157, 4.75927787377057189179851769076, 6.01828820532158172226743560416, 6.64078503298745854384165795732, 7.10553704569394816009048973191, 8.069911760640105394766209541968

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