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Properties

Label 2-1399-1399.1398-c0-0-14
Degree $2$
Conductor $1399$
Sign $1$
Analytic cond. $0.698191$
Root an. cond. $0.835578$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.53·2-s + 1.34·4-s + 1.53·5-s − 7-s + 0.532·8-s + 9-s + 2.34·10-s − 1.87·11-s − 1.53·14-s − 0.532·16-s + 1.53·18-s + 1.53·19-s + 2.06·20-s − 2.87·22-s − 23-s + 1.34·25-s − 1.34·28-s − 1.87·29-s − 1.34·32-s − 1.53·35-s + 1.34·36-s + 2·37-s + 2.34·38-s + 0.815·40-s − 1.87·41-s − 2.53·44-s + 1.53·45-s + ⋯
L(s)  = 1  + 1.53·2-s + 1.34·4-s + 1.53·5-s − 7-s + 0.532·8-s + 9-s + 2.34·10-s − 1.87·11-s − 1.53·14-s − 0.532·16-s + 1.53·18-s + 1.53·19-s + 2.06·20-s − 2.87·22-s − 23-s + 1.34·25-s − 1.34·28-s − 1.87·29-s − 1.34·32-s − 1.53·35-s + 1.34·36-s + 2·37-s + 2.34·38-s + 0.815·40-s − 1.87·41-s − 2.53·44-s + 1.53·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1399\)
Sign: $1$
Analytic conductor: \(0.698191\)
Root analytic conductor: \(0.835578\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1399} (1398, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1399,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.638433010\)
\(L(\frac12)\) \(\approx\) \(2.638433010\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1399 \( 1 - T \)
good2 \( 1 - 1.53T + T^{2} \)
3 \( 1 - T^{2} \)
5 \( 1 - 1.53T + T^{2} \)
7 \( 1 + T + T^{2} \)
11 \( 1 + 1.87T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - 1.53T + T^{2} \)
23 \( 1 + T + T^{2} \)
29 \( 1 + 1.87T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 2T + T^{2} \)
41 \( 1 + 1.87T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - 2T + T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 0.347T + T^{2} \)
79 \( 1 - 0.347T + T^{2} \)
83 \( 1 - 0.347T + T^{2} \)
89 \( 1 - 0.347T + T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.826199055948155339667691177078, −9.391792299018495697443129064086, −7.82137868188774127449798402128, −6.99121378451049749166723384384, −6.11017881766363102915930276149, −5.51916544941483688054507290524, −4.96438850065821880035329955178, −3.71561410187239841358929924722, −2.81253003133055267590828515228, −1.98344970866788644551150217601, 1.98344970866788644551150217601, 2.81253003133055267590828515228, 3.71561410187239841358929924722, 4.96438850065821880035329955178, 5.51916544941483688054507290524, 6.11017881766363102915930276149, 6.99121378451049749166723384384, 7.82137868188774127449798402128, 9.391792299018495697443129064086, 9.826199055948155339667691177078

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