Properties

Label 2-390-1.1-c1-0-8
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 $1$

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 − 12-s − 13-s + 15-s + 16-s − 6·17-s − 18-s − 20-s − 4·23-s + 24-s + 25-s + 26-s − 27-s − 10·29-s − 30-s − 32-s + 6·34-s + 36-s − 6·37-s + 39-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 − 0.288·12-s − 0.277·13-s + 0.258·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.223·20-s − 0.834·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 1.85·29-s − 0.182·30-s − 0.176·32-s + 1.02·34-s + 1/6·36-s − 0.986·37-s + 0.160·39-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 390,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 + T \)
13 \( 1 + T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−10.99019785051140937532808495638, −9.954657556925906330787615955302, −9.074224045302095177064838582083, −8.072251512020700899711976441689, −7.13489552716111364035097044906, −6.27139115246731968799115000690, −5.02996251545709838502394177698, −3.75949148792159340735853491565, −2.01716871449082204952154497689, 0, 2.01716871449082204952154497689, 3.75949148792159340735853491565, 5.02996251545709838502394177698, 6.27139115246731968799115000690, 7.13489552716111364035097044906, 8.072251512020700899711976441689, 9.074224045302095177064838582083, 9.954657556925906330787615955302, 10.99019785051140937532808495638

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