Properties

Label 2-1336-1.1-c1-0-22
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.49·3-s + 0.494·5-s + 2.18·7-s + 3.23·9-s + 3.30·11-s + 3.85·13-s + 1.23·15-s − 2.84·17-s − 0.107·19-s + 5.46·21-s − 0.225·23-s − 4.75·25-s + 0.582·27-s − 7.21·29-s − 1.87·31-s + 8.24·33-s + 1.08·35-s − 5.20·37-s + 9.63·39-s + 6.59·41-s − 6.49·43-s + 1.59·45-s + 9.83·47-s − 2.20·49-s − 7.11·51-s + 3.68·53-s + 1.63·55-s + ⋯
L(s)  = 1  + 1.44·3-s + 0.221·5-s + 0.827·7-s + 1.07·9-s + 0.995·11-s + 1.06·13-s + 0.318·15-s − 0.691·17-s − 0.0245·19-s + 1.19·21-s − 0.0470·23-s − 0.951·25-s + 0.112·27-s − 1.34·29-s − 0.336·31-s + 1.43·33-s + 0.183·35-s − 0.855·37-s + 1.54·39-s + 1.02·41-s − 0.990·43-s + 0.238·45-s + 1.43·47-s − 0.315·49-s − 0.996·51-s + 0.506·53-s + 0.220·55-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

\(L(1)\) \(\approx\) \(3.257755269\)
\(L(\frac12)\) \(\approx\) \(3.257755269\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.49T + 3T^{2} \)
5 \( 1 - 0.494T + 5T^{2} \)
7 \( 1 - 2.18T + 7T^{2} \)
11 \( 1 - 3.30T + 11T^{2} \)
13 \( 1 - 3.85T + 13T^{2} \)
17 \( 1 + 2.84T + 17T^{2} \)
19 \( 1 + 0.107T + 19T^{2} \)
23 \( 1 + 0.225T + 23T^{2} \)
29 \( 1 + 7.21T + 29T^{2} \)
31 \( 1 + 1.87T + 31T^{2} \)
37 \( 1 + 5.20T + 37T^{2} \)
41 \( 1 - 6.59T + 41T^{2} \)
43 \( 1 + 6.49T + 43T^{2} \)
47 \( 1 - 9.83T + 47T^{2} \)
53 \( 1 - 3.68T + 53T^{2} \)
59 \( 1 - 5.17T + 59T^{2} \)
61 \( 1 - 5.59T + 61T^{2} \)
67 \( 1 + 4.59T + 67T^{2} \)
71 \( 1 - 5.33T + 71T^{2} \)
73 \( 1 - 2.39T + 73T^{2} \)
79 \( 1 + 6.02T + 79T^{2} \)
83 \( 1 - 4.72T + 83T^{2} \)
89 \( 1 - 0.842T + 89T^{2} \)
97 \( 1 - 14.3T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.190567770123026426937136245351, −8.943298349022393508523261063261, −8.129007619860827307639223934677, −7.42752310936290210049396360254, −6.43301620810986064419396025926, −5.42046660929046056045582004602, −4.07907339783622369969991297380, −3.64874401017678139936554284305, −2.28053199297104851177852492938, −1.51788712502748915716106788013, 1.51788712502748915716106788013, 2.28053199297104851177852492938, 3.64874401017678139936554284305, 4.07907339783622369969991297380, 5.42046660929046056045582004602, 6.43301620810986064419396025926, 7.42752310936290210049396360254, 8.129007619860827307639223934677, 8.943298349022393508523261063261, 9.190567770123026426937136245351

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