Properties

Label 2-387-1.1-c7-0-103
Degree $2$
Conductor $387$
Sign $-1$
Analytic cond. $120.893$
Root an. cond. $10.9951$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7.80·2-s − 67.0·4-s + 402.·5-s − 356.·7-s − 1.52e3·8-s + 3.13e3·10-s − 338.·11-s − 1.90e3·13-s − 2.78e3·14-s − 3.30e3·16-s + 1.37e4·17-s + 3.23e4·19-s − 2.69e4·20-s − 2.64e3·22-s − 1.03e5·23-s + 8.34e4·25-s − 1.48e4·26-s + 2.39e4·28-s − 97.3·29-s − 2.91e3·31-s + 1.69e5·32-s + 1.07e5·34-s − 1.43e5·35-s − 2.80e5·37-s + 2.52e5·38-s − 6.12e5·40-s + 6.80e4·41-s + ⋯
L(s)  = 1  + 0.689·2-s − 0.523·4-s + 1.43·5-s − 0.393·7-s − 1.05·8-s + 0.992·10-s − 0.0767·11-s − 0.240·13-s − 0.271·14-s − 0.201·16-s + 0.676·17-s + 1.08·19-s − 0.753·20-s − 0.0529·22-s − 1.76·23-s + 1.06·25-s − 0.165·26-s + 0.205·28-s − 0.000741·29-s − 0.0176·31-s + 0.912·32-s + 0.467·34-s − 0.565·35-s − 0.909·37-s + 0.747·38-s − 1.51·40-s + 0.154·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(387\)    =    \(3^{2} \cdot 43\)
Sign: $-1$
Analytic conductor: \(120.893\)
Root analytic conductor: \(10.9951\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{387} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 387,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
43 \( 1 + 7.95e4T \)
good2 \( 1 - 7.80T + 128T^{2} \)
5 \( 1 - 402.T + 7.81e4T^{2} \)
7 \( 1 + 356.T + 8.23e5T^{2} \)
11 \( 1 + 338.T + 1.94e7T^{2} \)
13 \( 1 + 1.90e3T + 6.27e7T^{2} \)
17 \( 1 - 1.37e4T + 4.10e8T^{2} \)
19 \( 1 - 3.23e4T + 8.93e8T^{2} \)
23 \( 1 + 1.03e5T + 3.40e9T^{2} \)
29 \( 1 + 97.3T + 1.72e10T^{2} \)
31 \( 1 + 2.91e3T + 2.75e10T^{2} \)
37 \( 1 + 2.80e5T + 9.49e10T^{2} \)
41 \( 1 - 6.80e4T + 1.94e11T^{2} \)
47 \( 1 + 1.45e5T + 5.06e11T^{2} \)
53 \( 1 - 4.50e5T + 1.17e12T^{2} \)
59 \( 1 + 2.16e6T + 2.48e12T^{2} \)
61 \( 1 + 2.40e5T + 3.14e12T^{2} \)
67 \( 1 - 7.63e5T + 6.06e12T^{2} \)
71 \( 1 + 2.56e5T + 9.09e12T^{2} \)
73 \( 1 + 4.25e6T + 1.10e13T^{2} \)
79 \( 1 - 6.60e6T + 1.92e13T^{2} \)
83 \( 1 + 1.57e6T + 2.71e13T^{2} \)
89 \( 1 + 1.01e7T + 4.42e13T^{2} \)
97 \( 1 + 1.90e6T + 8.07e13T^{2} \)
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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

−9.765530027240102827726159160969, −9.032830092177735642463736395324, −7.82105955409046997952400381692, −6.42175141068396826650191260507, −5.71488701501446572937721468314, −5.01871782095475703463120668930, −3.72293935542943495376575807126, −2.70377658364808323281908884965, −1.46456176219286545739061419179, 0, 1.46456176219286545739061419179, 2.70377658364808323281908884965, 3.72293935542943495376575807126, 5.01871782095475703463120668930, 5.71488701501446572937721468314, 6.42175141068396826650191260507, 7.82105955409046997952400381692, 9.032830092177735642463736395324, 9.765530027240102827726159160969

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