Properties

Label 2-5390-1.1-c1-0-122
Degree $2$
Conductor $5390$
Sign $-1$
Analytic cond. $43.0393$
Root an. cond. $6.56043$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 2·3-s + 4-s + 5-s − 2·6-s − 8-s + 9-s − 10-s − 11-s + 2·12-s − 2·13-s + 2·15-s + 16-s + 6·17-s − 18-s − 2·19-s + 20-s + 22-s − 6·23-s − 2·24-s + 25-s + 2·26-s − 4·27-s − 2·30-s − 8·31-s − 32-s − 2·33-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.447·5-s − 0.816·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s + 0.577·12-s − 0.554·13-s + 0.516·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.458·19-s + 0.223·20-s + 0.213·22-s − 1.25·23-s − 0.408·24-s + 1/5·25-s + 0.392·26-s − 0.769·27-s − 0.365·30-s − 1.43·31-s − 0.176·32-s − 0.348·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5390\)    =    \(2 \cdot 5 \cdot 7^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(43.0393\)
Root analytic conductor: \(6.56043\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5390,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 \)
11 \( 1 + T \)
good3 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−8.100287354722351740118396870835, −7.35539584869822957604594549716, −6.60917017530382664296252335156, −5.67726847630877229411662708881, −5.03214215361775283568750803279, −3.67841618616134056250085124940, −3.20161008208660542164368410867, −2.18066836222259146215022225113, −1.65324664936885486756008330583, 0, 1.65324664936885486756008330583, 2.18066836222259146215022225113, 3.20161008208660542164368410867, 3.67841618616134056250085124940, 5.03214215361775283568750803279, 5.67726847630877229411662708881, 6.60917017530382664296252335156, 7.35539584869822957604594549716, 8.100287354722351740118396870835

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