Properties

Label 2-342-1.1-c9-0-55
Degree $2$
Conductor $342$
Sign $-1$
Analytic cond. $176.142$
Root an. cond. $13.2718$
Motivic weight $9$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 16·2-s + 256·4-s + 684·5-s + 9.14e3·7-s − 4.09e3·8-s − 1.09e4·10-s − 5.79e3·11-s − 1.79e5·13-s − 1.46e5·14-s + 6.55e4·16-s + 5.94e5·17-s + 1.30e5·19-s + 1.75e5·20-s + 9.26e4·22-s + 1.74e6·23-s − 1.48e6·25-s + 2.87e6·26-s + 2.34e6·28-s − 4.31e6·29-s + 1.60e5·31-s − 1.04e6·32-s − 9.50e6·34-s + 6.25e6·35-s − 2.19e7·37-s − 2.08e6·38-s − 2.80e6·40-s − 2.94e5·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.489·5-s + 1.44·7-s − 0.353·8-s − 0.346·10-s − 0.119·11-s − 1.74·13-s − 1.01·14-s + 1/4·16-s + 1.72·17-s + 0.229·19-s + 0.244·20-s + 0.0843·22-s + 1.30·23-s − 0.760·25-s + 1.23·26-s + 0.720·28-s − 1.13·29-s + 0.0311·31-s − 0.176·32-s − 1.21·34-s + 0.704·35-s − 1.92·37-s − 0.162·38-s − 0.173·40-s − 0.0162·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $-1$
Analytic conductor: \(176.142\)
Root analytic conductor: \(13.2718\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 342,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p^{4} T \)
3 \( 1 \)
19 \( 1 - p^{4} T \)
good5 \( 1 - 684 T + p^{9} T^{2} \)
7 \( 1 - 1307 p T + p^{9} T^{2} \)
11 \( 1 + 5790 T + p^{9} T^{2} \)
13 \( 1 + 13837 p T + p^{9} T^{2} \)
17 \( 1 - 594093 T + p^{9} T^{2} \)
23 \( 1 - 1744767 T + p^{9} T^{2} \)
29 \( 1 + 4314387 T + p^{9} T^{2} \)
31 \( 1 - 160232 T + p^{9} T^{2} \)
37 \( 1 + 21943090 T + p^{9} T^{2} \)
41 \( 1 + 294816 T + p^{9} T^{2} \)
43 \( 1 + 39393148 T + p^{9} T^{2} \)
47 \( 1 + 46596360 T + p^{9} T^{2} \)
53 \( 1 + 22121703 T + p^{9} T^{2} \)
59 \( 1 + 33070233 T + p^{9} T^{2} \)
61 \( 1 - 188535938 T + p^{9} T^{2} \)
67 \( 1 + 20769067 T + p^{9} T^{2} \)
71 \( 1 - 232299978 T + p^{9} T^{2} \)
73 \( 1 + 3022183 T + p^{9} T^{2} \)
79 \( 1 + 446379406 T + p^{9} T^{2} \)
83 \( 1 + 794022846 T + p^{9} T^{2} \)
89 \( 1 + 90999336 T + p^{9} T^{2} \)
97 \( 1 + 123974170 T + p^{9} T^{2} \)
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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

−9.698496549674461351891327520698, −8.520465144713392807442996220049, −7.69104812560962385631423318346, −7.03934618671628664335909150072, −5.42547721784552739347623875481, −4.99786974778475595398451503809, −3.28072028515980892315762027438, −2.03229054879918073889335982040, −1.34448136559822445809810865722, 0, 1.34448136559822445809810865722, 2.03229054879918073889335982040, 3.28072028515980892315762027438, 4.99786974778475595398451503809, 5.42547721784552739347623875481, 7.03934618671628664335909150072, 7.69104812560962385631423318346, 8.520465144713392807442996220049, 9.698496549674461351891327520698

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