Properties

Label 2-2736-1.1-c1-0-43
Degree $2$
Conductor $2736$
Sign $-1$
Analytic cond. $21.8470$
Root an. cond. $4.67408$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·7-s − 4·13-s − 6·17-s − 19-s − 6·23-s − 5·25-s − 6·29-s − 2·31-s − 4·37-s − 6·41-s + 4·43-s + 6·47-s + 9·49-s − 6·53-s − 12·59-s + 14·61-s − 8·67-s + 14·73-s + 10·79-s − 12·83-s + 6·89-s − 16·91-s − 10·97-s + 10·103-s + 12·107-s − 16·109-s + 18·113-s + ⋯
L(s)  = 1  + 1.51·7-s − 1.10·13-s − 1.45·17-s − 0.229·19-s − 1.25·23-s − 25-s − 1.11·29-s − 0.359·31-s − 0.657·37-s − 0.937·41-s + 0.609·43-s + 0.875·47-s + 9/7·49-s − 0.824·53-s − 1.56·59-s + 1.79·61-s − 0.977·67-s + 1.63·73-s + 1.12·79-s − 1.31·83-s + 0.635·89-s − 1.67·91-s − 1.01·97-s + 0.985·103-s + 1.16·107-s − 1.53·109-s + 1.69·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-1$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2736} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -1)\)

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 \( 1 \)
19 \( 1 + T \)
good5 \( 1 + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−8.359688629223917749842479932615, −7.72795819597559924442133345234, −7.10678703287277123493279316153, −6.07507983377375758209992822317, −5.21403289865941834256074312934, −4.54790713149621293279441185278, −3.81156610301665935267786910207, −2.27147129954073815802695201168, −1.81801100280978906223857531826, 0, 1.81801100280978906223857531826, 2.27147129954073815802695201168, 3.81156610301665935267786910207, 4.54790713149621293279441185278, 5.21403289865941834256074312934, 6.07507983377375758209992822317, 7.10678703287277123493279316153, 7.72795819597559924442133345234, 8.359688629223917749842479932615

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