Properties

Label 2-389-1.1-c1-0-29
Degree $2$
Conductor $389$
Sign $1$
Analytic cond. $3.10618$
Root an. cond. $1.76243$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s + 2·4-s − 3·5-s + 4·6-s − 5·7-s + 9-s + 6·10-s − 4·11-s − 4·12-s − 3·13-s + 10·14-s + 6·15-s − 4·16-s − 6·17-s − 2·18-s + 5·19-s − 6·20-s + 10·21-s + 8·22-s − 4·23-s + 4·25-s + 6·26-s + 4·27-s − 10·28-s − 6·29-s − 12·30-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 4-s − 1.34·5-s + 1.63·6-s − 1.88·7-s + 1/3·9-s + 1.89·10-s − 1.20·11-s − 1.15·12-s − 0.832·13-s + 2.67·14-s + 1.54·15-s − 16-s − 1.45·17-s − 0.471·18-s + 1.14·19-s − 1.34·20-s + 2.18·21-s + 1.70·22-s − 0.834·23-s + 4/5·25-s + 1.17·26-s + 0.769·27-s − 1.88·28-s − 1.11·29-s − 2.19·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(389\)
Sign: $1$
Analytic conductor: \(3.10618\)
Root analytic conductor: \(1.76243\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 389,\ (\ :1/2),\ 1)\)

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)$
bad389 \( 1 - T \)
good2 \( 1 + p T + p T^{2} \)
3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
7 \( 1 + 5 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 + 9 T + p T^{2} \)
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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

−10.35143331288149667136655977862, −9.633078802184913454726670089801, −8.633205244563326241602650564321, −7.47490749578543088938020099445, −6.98596665282868921801197833011, −5.79340263392836527147559896903, −4.41689608366525782922561294304, −2.87609907126046520176342609472, 0, 0, 2.87609907126046520176342609472, 4.41689608366525782922561294304, 5.79340263392836527147559896903, 6.98596665282868921801197833011, 7.47490749578543088938020099445, 8.633205244563326241602650564321, 9.633078802184913454726670089801, 10.35143331288149667136655977862

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