Properties

Label 2-2738-1.1-c1-0-40
Degree $2$
Conductor $2738$
Sign $-1$
Analytic cond. $21.8630$
Root an. cond. $4.67579$
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  − 2-s − 0.605·3-s + 4-s − 2.32·5-s + 0.605·6-s + 0.184·7-s − 8-s − 2.63·9-s + 2.32·10-s − 3.42·11-s − 0.605·12-s − 0.254·13-s − 0.184·14-s + 1.40·15-s + 16-s + 5.79·17-s + 2.63·18-s + 1.20·19-s − 2.32·20-s − 0.111·21-s + 3.42·22-s + 8.62·23-s + 0.605·24-s + 0.416·25-s + 0.254·26-s + 3.41·27-s + 0.184·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.349·3-s + 0.5·4-s − 1.04·5-s + 0.247·6-s + 0.0698·7-s − 0.353·8-s − 0.877·9-s + 0.735·10-s − 1.03·11-s − 0.174·12-s − 0.0705·13-s − 0.0493·14-s + 0.363·15-s + 0.250·16-s + 1.40·17-s + 0.620·18-s + 0.275·19-s − 0.520·20-s − 0.0244·21-s + 0.730·22-s + 1.79·23-s + 0.123·24-s + 0.0832·25-s + 0.0498·26-s + 0.656·27-s + 0.0349·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2738\)    =    \(2 \cdot 37^{2}\)
Sign: $-1$
Analytic conductor: \(21.8630\)
Root analytic conductor: \(4.67579\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2738,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
37 \( 1 \)
good3 \( 1 + 0.605T + 3T^{2} \)
5 \( 1 + 2.32T + 5T^{2} \)
7 \( 1 - 0.184T + 7T^{2} \)
11 \( 1 + 3.42T + 11T^{2} \)
13 \( 1 + 0.254T + 13T^{2} \)
17 \( 1 - 5.79T + 17T^{2} \)
19 \( 1 - 1.20T + 19T^{2} \)
23 \( 1 - 8.62T + 23T^{2} \)
29 \( 1 - 5.85T + 29T^{2} \)
31 \( 1 - 2.37T + 31T^{2} \)
41 \( 1 + 8.31T + 41T^{2} \)
43 \( 1 + 1.32T + 43T^{2} \)
47 \( 1 - 8.89T + 47T^{2} \)
53 \( 1 - 12.0T + 53T^{2} \)
59 \( 1 + 11.9T + 59T^{2} \)
61 \( 1 + 4.60T + 61T^{2} \)
67 \( 1 + 12.9T + 67T^{2} \)
71 \( 1 + 7.10T + 71T^{2} \)
73 \( 1 + 15.1T + 73T^{2} \)
79 \( 1 + 11.5T + 79T^{2} \)
83 \( 1 - 8.49T + 83T^{2} \)
89 \( 1 + 5.98T + 89T^{2} \)
97 \( 1 + 1.35T + 97T^{2} \)
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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

−8.485601991423656488382183824907, −7.64000268145194295285902294582, −7.30424598206911971694415605115, −6.17561060885764703812061157567, −5.37346456253015183993932619671, −4.64959258570563124737367215568, −3.24855623941275165023668687099, −2.82884689971878969394594553948, −1.12724729020232269592666560025, 0, 1.12724729020232269592666560025, 2.82884689971878969394594553948, 3.24855623941275165023668687099, 4.64959258570563124737367215568, 5.37346456253015183993932619671, 6.17561060885764703812061157567, 7.30424598206911971694415605115, 7.64000268145194295285902294582, 8.485601991423656488382183824907

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