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Properties

Label 2-510e2-1.1-c1-0-34
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  − 7-s − 5·11-s − 4·13-s + 19-s + 9·29-s + 6·31-s − 3·37-s + 5·41-s − 2·43-s + 9·47-s − 6·49-s + 3·53-s + 6·59-s + 14·67-s − 8·71-s + 7·73-s + 5·77-s + 6·79-s + 12·83-s + 6·89-s + 4·91-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.377·7-s − 1.50·11-s − 1.10·13-s + 0.229·19-s + 1.67·29-s + 1.07·31-s − 0.493·37-s + 0.780·41-s − 0.304·43-s + 1.31·47-s − 6/7·49-s + 0.412·53-s + 0.781·59-s + 1.71·67-s − 0.949·71-s + 0.819·73-s + 0.569·77-s + 0.675·79-s + 1.31·83-s + 0.635·89-s + 0.419·91-s − 0.203·97-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

\(L(1)\) \(\approx\) \(2.048645372\)
\(L(\frac12)\) \(\approx\) \(2.048645372\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−12.76468650882953, −12.32314401319313, −12.04205048492223, −11.50767445257042, −10.81319146568780, −10.47767461555871, −10.03916166978159, −9.733816466724292, −9.152066185432608, −8.521427640511701, −8.126844123287403, −7.658091502167652, −7.254370430798356, −6.595784500661694, −6.309364419995716, −5.479540281028239, −5.176836070499861, −4.714269886060658, −4.184147330173980, −3.410933198879987, −2.862487218553970, −2.474123234863065, −2.005078709440828, −0.8856040059767614, −0.4782318854711585, 0.4782318854711585, 0.8856040059767614, 2.005078709440828, 2.474123234863065, 2.862487218553970, 3.410933198879987, 4.184147330173980, 4.714269886060658, 5.176836070499861, 5.479540281028239, 6.309364419995716, 6.595784500661694, 7.254370430798356, 7.658091502167652, 8.126844123287403, 8.521427640511701, 9.152066185432608, 9.733816466724292, 10.03916166978159, 10.47767461555871, 10.81319146568780, 11.50767445257042, 12.04205048492223, 12.32314401319313, 12.76468650882953

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