Properties

Label 2-510e2-1.1-c1-0-26
Degree $2$
Conductor $260100$
Sign $1$
Analytic cond. $2076.90$
Root an. cond. $45.5731$
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  − 4·7-s + 5·13-s − 19-s − 7·31-s + 11·37-s + 5·43-s + 9·49-s + 14·61-s − 16·67-s − 10·73-s − 4·79-s − 20·91-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 1.51·7-s + 1.38·13-s − 0.229·19-s − 1.25·31-s + 1.80·37-s + 0.762·43-s + 9/7·49-s + 1.79·61-s − 1.95·67-s − 1.17·73-s − 0.450·79-s − 2.09·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(2076.90\)
Root analytic conductor: \(45.5731\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 260100,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.771920820\)
\(L(\frac12)\) \(\approx\) \(1.771920820\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
17 \( 1 \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−12.89106552150639, −12.59654691721510, −11.76250712911570, −11.52335717268091, −10.93661315426660, −10.42924730532819, −10.14692558821259, −9.434662275622893, −9.168906124301397, −8.772600849733712, −8.165003291962449, −7.612684024501994, −7.111714629348313, −6.604814279492519, −6.082056693107612, −5.877419699307997, −5.314912273465133, −4.400857889736928, −4.057963185958687, −3.508480262635310, −3.049792242171909, −2.513906364404203, −1.770748977230878, −1.030365211715883, −0.4076774511828008, 0.4076774511828008, 1.030365211715883, 1.770748977230878, 2.513906364404203, 3.049792242171909, 3.508480262635310, 4.057963185958687, 4.400857889736928, 5.314912273465133, 5.877419699307997, 6.082056693107612, 6.604814279492519, 7.111714629348313, 7.612684024501994, 8.165003291962449, 8.772600849733712, 9.168906124301397, 9.434662275622893, 10.14692558821259, 10.42924730532819, 10.93661315426660, 11.52335717268091, 11.76250712911570, 12.59654691721510, 12.89106552150639

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