Properties

Label 2-2738-1.1-c1-0-24
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2.73·3-s + 4-s − 1.73·5-s + 2.73·6-s + 4·7-s − 8-s + 4.46·9-s + 1.73·10-s − 1.26·11-s − 2.73·12-s + 6·13-s − 4·14-s + 4.73·15-s + 16-s + 1.73·17-s − 4.46·18-s + 4.73·19-s − 1.73·20-s − 10.9·21-s + 1.26·22-s + 4.73·23-s + 2.73·24-s − 2.00·25-s − 6·26-s − 3.99·27-s + 4·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.57·3-s + 0.5·4-s − 0.774·5-s + 1.11·6-s + 1.51·7-s − 0.353·8-s + 1.48·9-s + 0.547·10-s − 0.382·11-s − 0.788·12-s + 1.66·13-s − 1.06·14-s + 1.22·15-s + 0.250·16-s + 0.420·17-s − 1.05·18-s + 1.08·19-s − 0.387·20-s − 2.38·21-s + 0.270·22-s + 0.986·23-s + 0.557·24-s − 0.400·25-s − 1.17·26-s − 0.769·27-s + 0.755·28-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: \(0\)
Selberg data: \((2,\ 2738,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9120788720\)
\(L(\frac12)\) \(\approx\) \(0.9120788720\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.73T + 3T^{2} \)
5 \( 1 + 1.73T + 5T^{2} \)
7 \( 1 - 4T + 7T^{2} \)
11 \( 1 + 1.26T + 11T^{2} \)
13 \( 1 - 6T + 13T^{2} \)
17 \( 1 - 1.73T + 17T^{2} \)
19 \( 1 - 4.73T + 19T^{2} \)
23 \( 1 - 4.73T + 23T^{2} \)
29 \( 1 - 7.73T + 29T^{2} \)
31 \( 1 - 4.73T + 31T^{2} \)
41 \( 1 - 6.46T + 41T^{2} \)
43 \( 1 - 2.53T + 43T^{2} \)
47 \( 1 + 5.66T + 47T^{2} \)
53 \( 1 + 9.46T + 53T^{2} \)
59 \( 1 + 9.46T + 59T^{2} \)
61 \( 1 + 2.66T + 61T^{2} \)
67 \( 1 - 4.19T + 67T^{2} \)
71 \( 1 - 9.46T + 71T^{2} \)
73 \( 1 + 8.39T + 73T^{2} \)
79 \( 1 + 2.19T + 79T^{2} \)
83 \( 1 + 5.66T + 83T^{2} \)
89 \( 1 - 5.19T + 89T^{2} \)
97 \( 1 + 5.19T + 97T^{2} \)
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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

−8.599488106621334819476138391860, −8.022600604152041363252689207165, −7.45761164871895140602240332432, −6.49846060521260869608387767596, −5.80110487608468610912328065791, −5.00043054811138623370139865911, −4.36253531476792157809360052384, −3.13037782551287186079151416117, −1.41394119797288541573371244453, −0.834346030079800130478197683735, 0.834346030079800130478197683735, 1.41394119797288541573371244453, 3.13037782551287186079151416117, 4.36253531476792157809360052384, 5.00043054811138623370139865911, 5.80110487608468610912328065791, 6.49846060521260869608387767596, 7.45761164871895140602240332432, 8.022600604152041363252689207165, 8.599488106621334819476138391860

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