Properties

Label 2-6930-1.1-c1-0-21
Degree $2$
Conductor $6930$
Sign $1$
Analytic cond. $55.3363$
Root an. cond. $7.43883$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s + 11-s + 2·13-s + 14-s + 16-s + 3.65·17-s + 2.82·19-s + 20-s − 22-s − 2.82·23-s + 25-s − 2·26-s − 28-s − 8.82·29-s + 5.65·31-s − 32-s − 3.65·34-s − 35-s + 8.82·37-s − 2.82·38-s − 40-s − 4.82·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 0.447·5-s − 0.377·7-s − 0.353·8-s − 0.316·10-s + 0.301·11-s + 0.554·13-s + 0.267·14-s + 0.250·16-s + 0.886·17-s + 0.648·19-s + 0.223·20-s − 0.213·22-s − 0.589·23-s + 0.200·25-s − 0.392·26-s − 0.188·28-s − 1.63·29-s + 1.01·31-s − 0.176·32-s − 0.627·34-s − 0.169·35-s + 1.45·37-s − 0.458·38-s − 0.158·40-s − 0.754·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6930\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(55.3363\)
Root analytic conductor: \(7.43883\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6930,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.627463752\)
\(L(\frac12)\) \(\approx\) \(1.627463752\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 - T \)
good13 \( 1 - 2T + 13T^{2} \)
17 \( 1 - 3.65T + 17T^{2} \)
19 \( 1 - 2.82T + 19T^{2} \)
23 \( 1 + 2.82T + 23T^{2} \)
29 \( 1 + 8.82T + 29T^{2} \)
31 \( 1 - 5.65T + 31T^{2} \)
37 \( 1 - 8.82T + 37T^{2} \)
41 \( 1 + 4.82T + 41T^{2} \)
43 \( 1 - 1.65T + 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + 6.48T + 53T^{2} \)
59 \( 1 - 8T + 59T^{2} \)
61 \( 1 - 13.3T + 61T^{2} \)
67 \( 1 + 7.31T + 67T^{2} \)
71 \( 1 + 5.65T + 71T^{2} \)
73 \( 1 + 3.65T + 73T^{2} \)
79 \( 1 + 10.8T + 79T^{2} \)
83 \( 1 - 13.6T + 83T^{2} \)
89 \( 1 - 0.343T + 89T^{2} \)
97 \( 1 - 4.82T + 97T^{2} \)
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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

−7.958396207286525661007157354510, −7.37182532142529838037535128381, −6.59960604441439169607713175404, −5.89395603376242765595164866473, −5.44264740363302967203712321731, −4.23329283308185361339478924468, −3.45211045655002986373715699057, −2.62640663004094932202407881308, −1.64138778229613581179116197128, −0.75523982962365721079831247592, 0.75523982962365721079831247592, 1.64138778229613581179116197128, 2.62640663004094932202407881308, 3.45211045655002986373715699057, 4.23329283308185361339478924468, 5.44264740363302967203712321731, 5.89395603376242765595164866473, 6.59960604441439169607713175404, 7.37182532142529838037535128381, 7.958396207286525661007157354510

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