Properties

Label 2-1336-1.1-c1-0-4
Degree $2$
Conductor $1336$
Sign $1$
Analytic cond. $10.6680$
Root an. cond. $3.26619$
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  − 0.830·3-s − 1.03·5-s − 1.08·7-s − 2.31·9-s − 5.19·11-s − 0.733·13-s + 0.859·15-s + 5.46·17-s + 7.16·19-s + 0.898·21-s − 3.63·23-s − 3.93·25-s + 4.41·27-s + 7.44·29-s + 0.434·31-s + 4.31·33-s + 1.11·35-s + 6.82·37-s + 0.609·39-s + 9.91·41-s − 8.12·43-s + 2.38·45-s + 9.13·47-s − 5.82·49-s − 4.53·51-s + 7.48·53-s + 5.37·55-s + ⋯
L(s)  = 1  − 0.479·3-s − 0.462·5-s − 0.408·7-s − 0.770·9-s − 1.56·11-s − 0.203·13-s + 0.221·15-s + 1.32·17-s + 1.64·19-s + 0.196·21-s − 0.758·23-s − 0.786·25-s + 0.848·27-s + 1.38·29-s + 0.0779·31-s + 0.751·33-s + 0.189·35-s + 1.12·37-s + 0.0975·39-s + 1.54·41-s − 1.23·43-s + 0.356·45-s + 1.33·47-s − 0.832·49-s − 0.635·51-s + 1.02·53-s + 0.724·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1336\)    =    \(2^{3} \cdot 167\)
Sign: $1$
Analytic conductor: \(10.6680\)
Root analytic conductor: \(3.26619\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1336,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8945017213\)
\(L(\frac12)\) \(\approx\) \(0.8945017213\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
167 \( 1 + T \)
good3 \( 1 + 0.830T + 3T^{2} \)
5 \( 1 + 1.03T + 5T^{2} \)
7 \( 1 + 1.08T + 7T^{2} \)
11 \( 1 + 5.19T + 11T^{2} \)
13 \( 1 + 0.733T + 13T^{2} \)
17 \( 1 - 5.46T + 17T^{2} \)
19 \( 1 - 7.16T + 19T^{2} \)
23 \( 1 + 3.63T + 23T^{2} \)
29 \( 1 - 7.44T + 29T^{2} \)
31 \( 1 - 0.434T + 31T^{2} \)
37 \( 1 - 6.82T + 37T^{2} \)
41 \( 1 - 9.91T + 41T^{2} \)
43 \( 1 + 8.12T + 43T^{2} \)
47 \( 1 - 9.13T + 47T^{2} \)
53 \( 1 - 7.48T + 53T^{2} \)
59 \( 1 - 6.12T + 59T^{2} \)
61 \( 1 + 3.72T + 61T^{2} \)
67 \( 1 + 1.15T + 67T^{2} \)
71 \( 1 + 11.2T + 71T^{2} \)
73 \( 1 + 5.23T + 73T^{2} \)
79 \( 1 + 10.0T + 79T^{2} \)
83 \( 1 - 3.84T + 83T^{2} \)
89 \( 1 + 2.20T + 89T^{2} \)
97 \( 1 - 19.2T + 97T^{2} \)
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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

−9.977339948744352304325902690493, −8.713667802718933534173804227117, −7.78434264170895565160671859439, −7.46923217765195630571760636182, −6.02952041172400648652769316462, −5.57251261954866997604225786468, −4.67387893637278774885868869946, −3.34018244963435993968909484066, −2.63842526533332100693388387889, −0.68326119589542720648670334673, 0.68326119589542720648670334673, 2.63842526533332100693388387889, 3.34018244963435993968909484066, 4.67387893637278774885868869946, 5.57251261954866997604225786468, 6.02952041172400648652769316462, 7.46923217765195630571760636182, 7.78434264170895565160671859439, 8.713667802718933534173804227117, 9.977339948744352304325902690493

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