Properties

Label 2-380926-1.1-c1-0-24
Degree $2$
Conductor $380926$
Sign $-1$
Analytic cond. $3041.70$
Root an. cond. $55.1516$
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 − 2·6-s + 8-s + 9-s − 4·11-s − 2·12-s + 16-s − 6·17-s + 18-s − 6·19-s − 4·22-s − 23-s − 2·24-s − 5·25-s + 4·27-s + 10·29-s + 4·31-s + 32-s + 8·33-s − 6·34-s + 36-s + 2·37-s − 6·38-s − 10·41-s − 4·43-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s + 0.353·8-s + 1/3·9-s − 1.20·11-s − 0.577·12-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 1.37·19-s − 0.852·22-s − 0.208·23-s − 0.408·24-s − 25-s + 0.769·27-s + 1.85·29-s + 0.718·31-s + 0.176·32-s + 1.39·33-s − 1.02·34-s + 1/6·36-s + 0.328·37-s − 0.973·38-s − 1.56·41-s − 0.609·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380926 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380926 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380926\)    =    \(2 \cdot 7^{2} \cdot 13^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(3041.70\)
Root analytic conductor: \(55.1516\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{380926} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 380926,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 \)
13 \( 1 \)
23 \( 1 + T \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−12.73186724503055, −12.26925878666311, −11.76167561205255, −11.39541093097640, −11.03915167398632, −10.50283176204958, −10.18859091022108, −9.929050672506633, −8.910140227383352, −8.505344065618858, −8.221086206191656, −7.535284991631165, −6.926257197020727, −6.561040194878795, −6.161636211649798, −5.773574059255536, −5.217454072565674, −4.654796231175254, −4.503898069987110, −3.933170797843569, −3.058883914021522, −2.614748634392578, −2.153074698998907, −1.463612091872524, −0.5262039224858690, 0, 0.5262039224858690, 1.463612091872524, 2.153074698998907, 2.614748634392578, 3.058883914021522, 3.933170797843569, 4.503898069987110, 4.654796231175254, 5.217454072565674, 5.773574059255536, 6.161636211649798, 6.561040194878795, 6.926257197020727, 7.535284991631165, 8.221086206191656, 8.505344065618858, 8.910140227383352, 9.929050672506633, 10.18859091022108, 10.50283176204958, 11.03915167398632, 11.39541093097640, 11.76167561205255, 12.26925878666311, 12.73186724503055

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