Properties

Label 2-1386-1.1-c1-0-3
Degree $2$
Conductor $1386$
Sign $1$
Analytic cond. $11.0672$
Root an. cond. $3.32675$
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 + 4-s − 1.09·5-s + 7-s − 8-s + 1.09·10-s − 11-s + 6.80·13-s − 14-s + 16-s − 1.09·17-s − 3.89·19-s − 1.09·20-s + 22-s + 6.99·23-s − 3.80·25-s − 6.80·26-s + 28-s − 4.80·29-s − 1.27·31-s − 32-s + 1.09·34-s − 1.09·35-s + 4.18·37-s + 3.89·38-s + 1.09·40-s + 1.09·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.488·5-s + 0.377·7-s − 0.353·8-s + 0.345·10-s − 0.301·11-s + 1.88·13-s − 0.267·14-s + 0.250·16-s − 0.264·17-s − 0.894·19-s − 0.244·20-s + 0.213·22-s + 1.45·23-s − 0.761·25-s − 1.33·26-s + 0.188·28-s − 0.892·29-s − 0.229·31-s − 0.176·32-s + 0.187·34-s − 0.184·35-s + 0.687·37-s + 0.632·38-s + 0.172·40-s + 0.170·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.164071760\)
\(L(\frac12)\) \(\approx\) \(1.164071760\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
7 \( 1 - T \)
11 \( 1 + T \)
good5 \( 1 + 1.09T + 5T^{2} \)
13 \( 1 - 6.80T + 13T^{2} \)
17 \( 1 + 1.09T + 17T^{2} \)
19 \( 1 + 3.89T + 19T^{2} \)
23 \( 1 - 6.99T + 23T^{2} \)
29 \( 1 + 4.80T + 29T^{2} \)
31 \( 1 + 1.27T + 31T^{2} \)
37 \( 1 - 4.18T + 37T^{2} \)
41 \( 1 - 1.09T + 41T^{2} \)
43 \( 1 - 10.6T + 43T^{2} \)
47 \( 1 - 9.71T + 47T^{2} \)
53 \( 1 + 3.81T + 53T^{2} \)
59 \( 1 + 11.7T + 59T^{2} \)
61 \( 1 - 11.1T + 61T^{2} \)
67 \( 1 - 8.99T + 67T^{2} \)
71 \( 1 - 9.17T + 71T^{2} \)
73 \( 1 + 5.09T + 73T^{2} \)
79 \( 1 + 4.99T + 79T^{2} \)
83 \( 1 - 7.09T + 83T^{2} \)
89 \( 1 - 12.9T + 89T^{2} \)
97 \( 1 + 7.61T + 97T^{2} \)
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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

−9.314661444517123620766509668476, −8.793839906585969609406970472948, −8.043913396954558549893929816633, −7.36429574238662726859879157905, −6.36516766891233568471330292268, −5.62023244409812773786000646506, −4.32517181651641376904048294181, −3.48158755369553049833272524310, −2.17156949132925446450463602492, −0.890362842180685868572167976019, 0.890362842180685868572167976019, 2.17156949132925446450463602492, 3.48158755369553049833272524310, 4.32517181651641376904048294181, 5.62023244409812773786000646506, 6.36516766891233568471330292268, 7.36429574238662726859879157905, 8.043913396954558549893929816633, 8.793839906585969609406970472948, 9.314661444517123620766509668476

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