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Properties

Label 2-154800-1.1-c1-0-88
Degree $2$
Conductor $154800$
Sign $1$
Analytic cond. $1236.08$
Root an. cond. $35.1579$
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 + 4·11-s − 2·13-s + 6·17-s + 6·19-s + 6·23-s + 2·29-s + 4·31-s − 8·37-s − 8·41-s − 43-s + 6·47-s + 9·49-s + 6·53-s − 10·61-s + 12·67-s + 16·71-s − 16·73-s + 16·77-s + 4·79-s − 6·83-s − 2·89-s − 8·91-s + 18·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  + 1.51·7-s + 1.20·11-s − 0.554·13-s + 1.45·17-s + 1.37·19-s + 1.25·23-s + 0.371·29-s + 0.718·31-s − 1.31·37-s − 1.24·41-s − 0.152·43-s + 0.875·47-s + 9/7·49-s + 0.824·53-s − 1.28·61-s + 1.46·67-s + 1.89·71-s − 1.87·73-s + 1.82·77-s + 0.450·79-s − 0.658·83-s − 0.211·89-s − 0.838·91-s + 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 154800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(154800\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 43\)
Sign: $1$
Analytic conductor: \(1236.08\)
Root analytic conductor: \(35.1579\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 154800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.145596488\)
\(L(\frac12)\) \(\approx\) \(5.145596488\)
\(L(\frac{3}{2})\) 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 \( 1 \)
43 \( 1 + T \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
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 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−13.61245887751308, −12.66449744722390, −12.14456429683497, −11.89575588895990, −11.55282980461239, −11.03702245160444, −10.37475635902340, −10.05002834854241, −9.416690397398401, −8.973597335789016, −8.456078561176170, −7.978500939168292, −7.461752416437523, −7.044373012638986, −6.580065584662333, −5.698064921201692, −5.298781580182717, −4.920805448391429, −4.399240531027841, −3.606664770012368, −3.249333229588522, −2.496179235007666, −1.664972678710257, −1.235954087756647, −0.7539365542236988, 0.7539365542236988, 1.235954087756647, 1.664972678710257, 2.496179235007666, 3.249333229588522, 3.606664770012368, 4.399240531027841, 4.920805448391429, 5.298781580182717, 5.698064921201692, 6.580065584662333, 7.044373012638986, 7.461752416437523, 7.978500939168292, 8.456078561176170, 8.973597335789016, 9.416690397398401, 10.05002834854241, 10.37475635902340, 11.03702245160444, 11.55282980461239, 11.89575588895990, 12.14456429683497, 12.66449744722390, 13.61245887751308

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