Properties

Label 2-390-1.1-c1-0-0
Degree $2$
Conductor $390$
Sign $1$
Analytic cond. $3.11416$
Root an. cond. $1.76469$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 5-s + 6-s − 2·7-s − 8-s + 9-s − 10-s + 4·11-s − 12-s − 13-s + 2·14-s − 15-s + 16-s + 4·17-s − 18-s − 2·19-s + 20-s + 2·21-s − 4·22-s + 2·23-s + 24-s + 25-s + 26-s − 27-s − 2·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s − 0.288·12-s − 0.277·13-s + 0.534·14-s − 0.258·15-s + 1/4·16-s + 0.970·17-s − 0.235·18-s − 0.458·19-s + 0.223·20-s + 0.436·21-s − 0.852·22-s + 0.417·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 0.377·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(390\)    =    \(2 \cdot 3 \cdot 5 \cdot 13\)
Sign: $1$
Analytic conductor: \(3.11416\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{390} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8835342653\)
\(L(\frac12)\) \(\approx\) \(0.8835342653\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 - T \)
13 \( 1 + T \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−11.21702121751910391674584411904, −10.21018500810305834142688498374, −9.620976763027980482767638198164, −8.772194400278408675663060319707, −7.50072014239365599576466271951, −6.49330656599515595141207552224, −5.92218145052084868507888987744, −4.39848825034532417015468798767, −2.87565984705891767205138792492, −1.10731363493759878565508314296, 1.10731363493759878565508314296, 2.87565984705891767205138792492, 4.39848825034532417015468798767, 5.92218145052084868507888987744, 6.49330656599515595141207552224, 7.50072014239365599576466271951, 8.772194400278408675663060319707, 9.620976763027980482767638198164, 10.21018500810305834142688498374, 11.21702121751910391674584411904

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