Properties

Label 2-354-1.1-c3-0-27
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 − 13.6·5-s + 6·6-s + 4.72·7-s + 8·8-s + 9·9-s − 27.2·10-s − 58.0·11-s + 12·12-s − 87.0·13-s + 9.44·14-s − 40.8·15-s + 16·16-s − 17.9·17-s + 18·18-s − 29.3·19-s − 54.4·20-s + 14.1·21-s − 116.·22-s + 172.·23-s + 24·24-s + 60.1·25-s − 174.·26-s + 27·27-s + 18.8·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.21·5-s + 0.408·6-s + 0.255·7-s + 0.353·8-s + 0.333·9-s − 0.860·10-s − 1.59·11-s + 0.288·12-s − 1.85·13-s + 0.180·14-s − 0.702·15-s + 0.250·16-s − 0.255·17-s + 0.235·18-s − 0.353·19-s − 0.608·20-s + 0.147·21-s − 1.12·22-s + 1.56·23-s + 0.204·24-s + 0.480·25-s − 1.31·26-s + 0.192·27-s + 0.127·28-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 \( 1 + 13.6T + 125T^{2} \)
7 \( 1 - 4.72T + 343T^{2} \)
11 \( 1 + 58.0T + 1.33e3T^{2} \)
13 \( 1 + 87.0T + 2.19e3T^{2} \)
17 \( 1 + 17.9T + 4.91e3T^{2} \)
19 \( 1 + 29.3T + 6.85e3T^{2} \)
23 \( 1 - 172.T + 1.21e4T^{2} \)
29 \( 1 - 41.9T + 2.43e4T^{2} \)
31 \( 1 + 287.T + 2.97e4T^{2} \)
37 \( 1 + 22.1T + 5.06e4T^{2} \)
41 \( 1 - 321.T + 6.89e4T^{2} \)
43 \( 1 + 423.T + 7.95e4T^{2} \)
47 \( 1 + 7.95T + 1.03e5T^{2} \)
53 \( 1 + 177.T + 1.48e5T^{2} \)
61 \( 1 - 217.T + 2.26e5T^{2} \)
67 \( 1 - 561.T + 3.00e5T^{2} \)
71 \( 1 - 572.T + 3.57e5T^{2} \)
73 \( 1 + 242.T + 3.89e5T^{2} \)
79 \( 1 + 587.T + 4.93e5T^{2} \)
83 \( 1 + 887.T + 5.71e5T^{2} \)
89 \( 1 + 428.T + 7.04e5T^{2} \)
97 \( 1 - 1.05e3T + 9.12e5T^{2} \)
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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

−10.80835653439212612808062116554, −9.745140274102151914456190099635, −8.428202999746882421239929726283, −7.57626833930540170745400011956, −7.08594403243481762508455341600, −5.22439331483275777344699861789, −4.59020574929205439176871432434, −3.28723322988448770571983106083, −2.33173747463694666456213607619, 0, 2.33173747463694666456213607619, 3.28723322988448770571983106083, 4.59020574929205439176871432434, 5.22439331483275777344699861789, 7.08594403243481762508455341600, 7.57626833930540170745400011956, 8.428202999746882421239929726283, 9.745140274102151914456190099635, 10.80835653439212612808062116554

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