Properties

Label 2-39-39.38-c0-0-0
Degree $2$
Conductor $39$
Sign $1$
Analytic cond. $0.0194635$
Root an. cond. $0.139511$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4-s + 9-s + 12-s − 13-s + 16-s − 25-s − 27-s − 36-s + 39-s + 2·43-s − 48-s + 49-s + 52-s − 2·61-s − 64-s + 75-s − 2·79-s + 81-s + 100-s + 2·103-s + 108-s − 117-s + ⋯
L(s)  = 1  − 3-s − 4-s + 9-s + 12-s − 13-s + 16-s − 25-s − 27-s − 36-s + 39-s + 2·43-s − 48-s + 49-s + 52-s − 2·61-s − 64-s + 75-s − 2·79-s + 81-s + 100-s + 2·103-s + 108-s − 117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $1$
Analytic conductor: \(0.0194635\)
Root analytic conductor: \(0.139511\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{39} (38, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2913537228\)
\(L(\frac12)\) \(\approx\) \(0.2913537228\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
13 \( 1 + T \)
good2 \( 1 + T^{2} \)
5 \( 1 + T^{2} \)
7 \( ( 1 - T )( 1 + T ) \)
11 \( 1 + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( 1 + T^{2} \)
43 \( ( 1 - T )^{2} \)
47 \( 1 + T^{2} \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( 1 + T^{2} \)
61 \( ( 1 + T )^{2} \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( 1 + T^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 + T )^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−16.89615994646558642421555820819, −15.51291484878802936574704349484, −14.14821592234820384952611063785, −12.89937159831719324582676854317, −11.93271353025467403728401266104, −10.42608525426608152464833032909, −9.334261916221688779645422148440, −7.54499009544086055036625670577, −5.72936210305730426417401616626, −4.39582175376675801531882680457, 4.39582175376675801531882680457, 5.72936210305730426417401616626, 7.54499009544086055036625670577, 9.334261916221688779645422148440, 10.42608525426608152464833032909, 11.93271353025467403728401266104, 12.89937159831719324582676854317, 14.14821592234820384952611063785, 15.51291484878802936574704349484, 16.89615994646558642421555820819

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