Properties

Label 2-342-1.1-c3-0-9
Degree $2$
Conductor $342$
Sign $1$
Analytic cond. $20.1786$
Root an. cond. $4.49206$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 4·4-s + 7·5-s − 15·7-s + 8·8-s + 14·10-s + 49·11-s + 14·13-s − 30·14-s + 16·16-s + 33·17-s − 19·19-s + 28·20-s + 98·22-s + 148·23-s − 76·25-s + 28·26-s − 60·28-s + 278·29-s + 94·31-s + 32·32-s + 66·34-s − 105·35-s + 160·37-s − 38·38-s + 56·40-s − 400·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.626·5-s − 0.809·7-s + 0.353·8-s + 0.442·10-s + 1.34·11-s + 0.298·13-s − 0.572·14-s + 1/4·16-s + 0.470·17-s − 0.229·19-s + 0.313·20-s + 0.949·22-s + 1.34·23-s − 0.607·25-s + 0.211·26-s − 0.404·28-s + 1.78·29-s + 0.544·31-s + 0.176·32-s + 0.332·34-s − 0.507·35-s + 0.710·37-s − 0.162·38-s + 0.221·40-s − 1.52·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(20.1786\)
Root analytic conductor: \(4.49206\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.348487041\)
\(L(\frac12)\) \(\approx\) \(3.348487041\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p T \)
3 \( 1 \)
19 \( 1 + p T \)
good5 \( 1 - 7 T + p^{3} T^{2} \)
7 \( 1 + 15 T + p^{3} T^{2} \)
11 \( 1 - 49 T + p^{3} T^{2} \)
13 \( 1 - 14 T + p^{3} T^{2} \)
17 \( 1 - 33 T + p^{3} T^{2} \)
23 \( 1 - 148 T + p^{3} T^{2} \)
29 \( 1 - 278 T + p^{3} T^{2} \)
31 \( 1 - 94 T + p^{3} T^{2} \)
37 \( 1 - 160 T + p^{3} T^{2} \)
41 \( 1 + 400 T + p^{3} T^{2} \)
43 \( 1 - 73 T + p^{3} T^{2} \)
47 \( 1 + 173 T + p^{3} T^{2} \)
53 \( 1 + 170 T + p^{3} T^{2} \)
59 \( 1 - 12 T + p^{3} T^{2} \)
61 \( 1 - 419 T + p^{3} T^{2} \)
67 \( 1 - 444 T + p^{3} T^{2} \)
71 \( 1 - 952 T + p^{3} T^{2} \)
73 \( 1 + 27 T + p^{3} T^{2} \)
79 \( 1 + 556 T + p^{3} T^{2} \)
83 \( 1 - 276 T + p^{3} T^{2} \)
89 \( 1 + 1386 T + p^{3} T^{2} \)
97 \( 1 - 130 T + p^{3} T^{2} \)
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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

−11.27761196108891979652189913624, −10.11298159246558372357794985179, −9.425843124485034893599299697981, −8.308064909252842707794338876768, −6.73068396926731449001752891010, −6.38648167744458584255270214204, −5.16415626272052223720040167244, −3.90984578055402605358485409444, −2.83886337591772895206143345325, −1.25375087497093894010927906662, 1.25375087497093894010927906662, 2.83886337591772895206143345325, 3.90984578055402605358485409444, 5.16415626272052223720040167244, 6.38648167744458584255270214204, 6.73068396926731449001752891010, 8.308064909252842707794338876768, 9.425843124485034893599299697981, 10.11298159246558372357794985179, 11.27761196108891979652189913624

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