Properties

Label 2-1344-1.1-c3-0-63
Degree $2$
Conductor $1344$
Sign $-1$
Analytic cond. $79.2985$
Root an. cond. $8.90497$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 4·5-s − 7·7-s + 9·9-s − 62·11-s + 62·13-s + 12·15-s + 84·17-s − 100·19-s − 21·21-s − 42·23-s − 109·25-s + 27·27-s + 10·29-s − 48·31-s − 186·33-s − 28·35-s + 246·37-s + 186·39-s − 248·41-s − 68·43-s + 36·45-s + 324·47-s + 49·49-s + 252·51-s − 258·53-s − 248·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.357·5-s − 0.377·7-s + 1/3·9-s − 1.69·11-s + 1.32·13-s + 0.206·15-s + 1.19·17-s − 1.20·19-s − 0.218·21-s − 0.380·23-s − 0.871·25-s + 0.192·27-s + 0.0640·29-s − 0.278·31-s − 0.981·33-s − 0.135·35-s + 1.09·37-s + 0.763·39-s − 0.944·41-s − 0.241·43-s + 0.119·45-s + 1.00·47-s + 1/7·49-s + 0.691·51-s − 0.668·53-s − 0.608·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-1$
Analytic conductor: \(79.2985\)
Root analytic conductor: \(8.90497\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1344,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - p T \)
7 \( 1 + p T \)
good5 \( 1 - 4 T + p^{3} T^{2} \)
11 \( 1 + 62 T + p^{3} T^{2} \)
13 \( 1 - 62 T + p^{3} T^{2} \)
17 \( 1 - 84 T + p^{3} T^{2} \)
19 \( 1 + 100 T + p^{3} T^{2} \)
23 \( 1 + 42 T + p^{3} T^{2} \)
29 \( 1 - 10 T + p^{3} T^{2} \)
31 \( 1 + 48 T + p^{3} T^{2} \)
37 \( 1 - 246 T + p^{3} T^{2} \)
41 \( 1 + 248 T + p^{3} T^{2} \)
43 \( 1 + 68 T + p^{3} T^{2} \)
47 \( 1 - 324 T + p^{3} T^{2} \)
53 \( 1 + 258 T + p^{3} T^{2} \)
59 \( 1 + 120 T + p^{3} T^{2} \)
61 \( 1 + 622 T + p^{3} T^{2} \)
67 \( 1 + 904 T + p^{3} T^{2} \)
71 \( 1 + 678 T + p^{3} T^{2} \)
73 \( 1 + 642 T + p^{3} T^{2} \)
79 \( 1 - 740 T + p^{3} T^{2} \)
83 \( 1 + 468 T + p^{3} T^{2} \)
89 \( 1 - 200 T + p^{3} T^{2} \)
97 \( 1 + 1266 T + p^{3} T^{2} \)
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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

−8.739250510367619291138401834871, −8.063011484403483794150764682539, −7.43816577747085928414009935745, −6.16524378370530187799461487610, −5.68537232990306670572870835930, −4.47359154573672067135348034971, −3.44719888659052754622041816101, −2.61299624017185976784268651282, −1.52536861439271447949090332570, 0, 1.52536861439271447949090332570, 2.61299624017185976784268651282, 3.44719888659052754622041816101, 4.47359154573672067135348034971, 5.68537232990306670572870835930, 6.16524378370530187799461487610, 7.43816577747085928414009935745, 8.063011484403483794150764682539, 8.739250510367619291138401834871

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