Properties

Label 2-207-1.1-c7-0-53
Degree $2$
Conductor $207$
Sign $-1$
Analytic cond. $64.6637$
Root an. cond. $8.04137$
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  − 9.40·2-s − 39.5·4-s + 482.·5-s + 1.33e3·7-s + 1.57e3·8-s − 4.53e3·10-s + 1.51e3·11-s − 1.42e4·13-s − 1.25e4·14-s − 9.75e3·16-s − 2.36e4·17-s − 4.11e4·19-s − 1.90e4·20-s − 1.42e4·22-s + 1.21e4·23-s + 1.54e5·25-s + 1.33e5·26-s − 5.26e4·28-s − 1.00e5·29-s − 2.49e5·31-s − 1.09e5·32-s + 2.22e5·34-s + 6.42e5·35-s + 1.84e5·37-s + 3.87e5·38-s + 7.60e5·40-s − 4.92e5·41-s + ⋯
L(s)  = 1  − 0.831·2-s − 0.309·4-s + 1.72·5-s + 1.46·7-s + 1.08·8-s − 1.43·10-s + 0.344·11-s − 1.79·13-s − 1.21·14-s − 0.595·16-s − 1.16·17-s − 1.37·19-s − 0.533·20-s − 0.286·22-s + 0.208·23-s + 1.98·25-s + 1.49·26-s − 0.453·28-s − 0.767·29-s − 1.50·31-s − 0.593·32-s + 0.971·34-s + 2.53·35-s + 0.598·37-s + 1.14·38-s + 1.87·40-s − 1.11·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(207\)    =    \(3^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(64.6637\)
Root analytic conductor: \(8.04137\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 207,\ (\ :7/2),\ -1)\)

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 \( 1 \)
23 \( 1 - 1.21e4T \)
good2 \( 1 + 9.40T + 128T^{2} \)
5 \( 1 - 482.T + 7.81e4T^{2} \)
7 \( 1 - 1.33e3T + 8.23e5T^{2} \)
11 \( 1 - 1.51e3T + 1.94e7T^{2} \)
13 \( 1 + 1.42e4T + 6.27e7T^{2} \)
17 \( 1 + 2.36e4T + 4.10e8T^{2} \)
19 \( 1 + 4.11e4T + 8.93e8T^{2} \)
29 \( 1 + 1.00e5T + 1.72e10T^{2} \)
31 \( 1 + 2.49e5T + 2.75e10T^{2} \)
37 \( 1 - 1.84e5T + 9.49e10T^{2} \)
41 \( 1 + 4.92e5T + 1.94e11T^{2} \)
43 \( 1 - 3.22e4T + 2.71e11T^{2} \)
47 \( 1 + 1.13e6T + 5.06e11T^{2} \)
53 \( 1 + 1.39e6T + 1.17e12T^{2} \)
59 \( 1 + 1.10e6T + 2.48e12T^{2} \)
61 \( 1 - 1.35e6T + 3.14e12T^{2} \)
67 \( 1 + 4.46e6T + 6.06e12T^{2} \)
71 \( 1 - 1.63e6T + 9.09e12T^{2} \)
73 \( 1 - 2.89e6T + 1.10e13T^{2} \)
79 \( 1 - 2.40e6T + 1.92e13T^{2} \)
83 \( 1 - 4.40e6T + 2.71e13T^{2} \)
89 \( 1 - 9.14e5T + 4.42e13T^{2} \)
97 \( 1 - 3.86e6T + 8.07e13T^{2} \)
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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

−10.43739207963765723747773127995, −9.490972216113907693836934800749, −8.913496895968612311163678902830, −7.81654895617964514102710307414, −6.65593514910742772213459183954, −5.19770976512646251904071122316, −4.59069436495358774575336966281, −2.09981533344842857365066448294, −1.69898338941572940873563171625, 0, 1.69898338941572940873563171625, 2.09981533344842857365066448294, 4.59069436495358774575336966281, 5.19770976512646251904071122316, 6.65593514910742772213459183954, 7.81654895617964514102710307414, 8.913496895968612311163678902830, 9.490972216113907693836934800749, 10.43739207963765723747773127995

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