Properties

Label 2-354-1.1-c3-0-23
Degree $2$
Conductor $354$
Sign $-1$
Analytic cond. $20.8866$
Root an. cond. $4.57019$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3·3-s + 4·4-s − 16.7·5-s − 6·6-s + 24.0·7-s + 8·8-s + 9·9-s − 33.4·10-s + 14.3·11-s − 12·12-s − 53.8·13-s + 48.0·14-s + 50.2·15-s + 16·16-s + 61.0·17-s + 18·18-s − 115.·19-s − 66.9·20-s − 72.0·21-s + 28.6·22-s − 188.·23-s − 24·24-s + 155.·25-s − 107.·26-s − 27·27-s + 96.0·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.49·5-s − 0.408·6-s + 1.29·7-s + 0.353·8-s + 0.333·9-s − 1.05·10-s + 0.392·11-s − 0.288·12-s − 1.14·13-s + 0.917·14-s + 0.864·15-s + 0.250·16-s + 0.871·17-s + 0.235·18-s − 1.39·19-s − 0.749·20-s − 0.748·21-s + 0.277·22-s − 1.70·23-s − 0.204·24-s + 1.24·25-s − 0.811·26-s − 0.192·27-s + 0.648·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(20.8866\)
Root analytic conductor: \(4.57019\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 354,\ (\ :3/2),\ -1)\)

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 \( 1 - 2T \)
3 \( 1 + 3T \)
59 \( 1 + 59T \)
good5 \( 1 + 16.7T + 125T^{2} \)
7 \( 1 - 24.0T + 343T^{2} \)
11 \( 1 - 14.3T + 1.33e3T^{2} \)
13 \( 1 + 53.8T + 2.19e3T^{2} \)
17 \( 1 - 61.0T + 4.91e3T^{2} \)
19 \( 1 + 115.T + 6.85e3T^{2} \)
23 \( 1 + 188.T + 1.21e4T^{2} \)
29 \( 1 + 216.T + 2.43e4T^{2} \)
31 \( 1 + 22.1T + 2.97e4T^{2} \)
37 \( 1 - 158.T + 5.06e4T^{2} \)
41 \( 1 + 380.T + 6.89e4T^{2} \)
43 \( 1 + 213.T + 7.95e4T^{2} \)
47 \( 1 + 426.T + 1.03e5T^{2} \)
53 \( 1 - 18.7T + 1.48e5T^{2} \)
61 \( 1 - 719.T + 2.26e5T^{2} \)
67 \( 1 - 107.T + 3.00e5T^{2} \)
71 \( 1 - 365.T + 3.57e5T^{2} \)
73 \( 1 + 341.T + 3.89e5T^{2} \)
79 \( 1 - 1.28e3T + 4.93e5T^{2} \)
83 \( 1 + 1.34e3T + 5.71e5T^{2} \)
89 \( 1 + 226.T + 7.04e5T^{2} \)
97 \( 1 - 664.T + 9.12e5T^{2} \)
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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

−11.00701986886811573549303841679, −9.954963867444308992168812416236, −8.233703924120726710124410060816, −7.76094783895474575534563955788, −6.73192157854847402695276045614, −5.37757823123400767544002951168, −4.49406296019897794126731729454, −3.74806354623674706393129482730, −1.88179002391441450122947694692, 0, 1.88179002391441450122947694692, 3.74806354623674706393129482730, 4.49406296019897794126731729454, 5.37757823123400767544002951168, 6.73192157854847402695276045614, 7.76094783895474575534563955788, 8.233703924120726710124410060816, 9.954963867444308992168812416236, 11.00701986886811573549303841679

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