Properties

Label 2-1950-1.1-c1-0-8
Degree $2$
Conductor $1950$
Sign $1$
Analytic cond. $15.5708$
Root an. cond. $3.94598$
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 − 6-s + 8-s + 9-s − 6·11-s − 12-s + 13-s + 16-s + 18-s + 6·19-s − 6·22-s + 6·23-s − 24-s + 26-s − 27-s + 2·29-s + 4·31-s + 32-s + 6·33-s + 36-s + 10·37-s + 6·38-s − 39-s − 6·41-s + 8·43-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 1.80·11-s − 0.288·12-s + 0.277·13-s + 1/4·16-s + 0.235·18-s + 1.37·19-s − 1.27·22-s + 1.25·23-s − 0.204·24-s + 0.196·26-s − 0.192·27-s + 0.371·29-s + 0.718·31-s + 0.176·32-s + 1.04·33-s + 1/6·36-s + 1.64·37-s + 0.973·38-s − 0.160·39-s − 0.937·41-s + 1.21·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1950} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.171628782\)
\(L(\frac12)\) \(\approx\) \(2.171628782\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−9.339196028527913011420132860253, −8.112828788704538337471455400291, −7.55791425352179681097047553232, −6.71090874096745784254140002420, −5.76561252871232307176139755211, −5.18274785794287017679754907894, −4.52351035255479102202576404027, −3.24832322846155318966060592343, −2.50735482069933531321125260906, −0.931742409450502916584918442470, 0.931742409450502916584918442470, 2.50735482069933531321125260906, 3.24832322846155318966060592343, 4.52351035255479102202576404027, 5.18274785794287017679754907894, 5.76561252871232307176139755211, 6.71090874096745784254140002420, 7.55791425352179681097047553232, 8.112828788704538337471455400291, 9.339196028527913011420132860253

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