Properties

Degree $2$
Conductor $390$
Sign $1$
Motivic weight $1$
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 + 8-s + 9-s + 10-s + 4·11-s − 12-s + 13-s − 15-s + 16-s − 6·17-s + 18-s + 4·19-s + 20-s + 4·22-s + 8·23-s − 24-s + 25-s + 26-s − 27-s + 6·29-s − 30-s − 8·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s − 0.288·12-s + 0.277·13-s − 0.258·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.852·22-s + 1.66·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s + 1.11·29-s − 0.182·30-s − 1.43·31-s + 0.176·32-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: $1$
Motivic weight: \(1\)
Character: $\chi_{390} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.959827583\)
\(L(\frac12)\) \(\approx\) \(1.959827583\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−19.57559704656104, −18.64973544906586, −17.60466206386819, −17.19844821470022, −16.21606666436362, −15.52250673514557, −14.58010706646970, −13.81365760807813, −13.04928947665351, −12.21361423981929, −11.32890074287175, −10.75872349499440, −9.514279277335677, −8.704161392497896, −7.035829833279218, −6.572676758264871, −5.431607793207443, −4.534967625163603, −3.266543285348448, −1.559771416907614, 1.559771416907614, 3.266543285348448, 4.534967625163603, 5.431607793207443, 6.572676758264871, 7.035829833279218, 8.704161392497896, 9.514279277335677, 10.75872349499440, 11.32890074287175, 12.21361423981929, 13.04928947665351, 13.81365760807813, 14.58010706646970, 15.52250673514557, 16.21606666436362, 17.19844821470022, 17.60466206386819, 18.64973544906586, 19.57559704656104

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