Properties

Label 2-2352-1.1-c3-0-87
Degree $2$
Conductor $2352$
Sign $-1$
Analytic cond. $138.772$
Root an. cond. $11.7801$
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  + 3·3-s − 10.5·5-s + 9·9-s − 34.7·11-s + 37.2·13-s − 31.6·15-s + 10.5·17-s − 58.5·19-s + 125.·23-s − 13.7·25-s + 27·27-s − 35.4·29-s + 291.·31-s − 104.·33-s − 259.·37-s + 111.·39-s + 338.·41-s − 6.80·43-s − 94.9·45-s + 250.·47-s + 31.6·51-s − 536.·53-s + 366.·55-s − 175.·57-s − 35.8·59-s − 57.7·61-s − 393.·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.943·5-s + 0.333·9-s − 0.952·11-s + 0.795·13-s − 0.544·15-s + 0.150·17-s − 0.707·19-s + 1.13·23-s − 0.109·25-s + 0.192·27-s − 0.226·29-s + 1.69·31-s − 0.549·33-s − 1.15·37-s + 0.459·39-s + 1.28·41-s − 0.0241·43-s − 0.314·45-s + 0.778·47-s + 0.0868·51-s − 1.39·53-s + 0.898·55-s − 0.408·57-s − 0.0791·59-s − 0.121·61-s − 0.750·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(138.772\)
Root analytic conductor: \(11.7801\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2352,\ (\ :3/2),\ -1)\)

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 \( 1 - 3T \)
7 \( 1 \)
good5 \( 1 + 10.5T + 125T^{2} \)
11 \( 1 + 34.7T + 1.33e3T^{2} \)
13 \( 1 - 37.2T + 2.19e3T^{2} \)
17 \( 1 - 10.5T + 4.91e3T^{2} \)
19 \( 1 + 58.5T + 6.85e3T^{2} \)
23 \( 1 - 125.T + 1.21e4T^{2} \)
29 \( 1 + 35.4T + 2.43e4T^{2} \)
31 \( 1 - 291.T + 2.97e4T^{2} \)
37 \( 1 + 259.T + 5.06e4T^{2} \)
41 \( 1 - 338.T + 6.89e4T^{2} \)
43 \( 1 + 6.80T + 7.95e4T^{2} \)
47 \( 1 - 250.T + 1.03e5T^{2} \)
53 \( 1 + 536.T + 1.48e5T^{2} \)
59 \( 1 + 35.8T + 2.05e5T^{2} \)
61 \( 1 + 57.7T + 2.26e5T^{2} \)
67 \( 1 + 481.T + 3.00e5T^{2} \)
71 \( 1 + 363.T + 3.57e5T^{2} \)
73 \( 1 + 581.T + 3.89e5T^{2} \)
79 \( 1 - 693.T + 4.93e5T^{2} \)
83 \( 1 - 1.33e3T + 5.71e5T^{2} \)
89 \( 1 - 353.T + 7.04e5T^{2} \)
97 \( 1 + 1.44e3T + 9.12e5T^{2} \)
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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

−8.183708164598938593441898074661, −7.69333961758841370526986745340, −6.86178402921126138828018829689, −5.94821738587100335265558533688, −4.88422898062411965712946029257, −4.13703419761613135581947811739, −3.28202969547634036390091411165, −2.50627674639791056357748268712, −1.19484605574209766719975417736, 0, 1.19484605574209766719975417736, 2.50627674639791056357748268712, 3.28202969547634036390091411165, 4.13703419761613135581947811739, 4.88422898062411965712946029257, 5.94821738587100335265558533688, 6.86178402921126138828018829689, 7.69333961758841370526986745340, 8.183708164598938593441898074661

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