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Properties

Label 2-10800-1.1-c1-0-30
Degree $2$
Conductor $10800$
Sign $1$
Analytic cond. $86.2384$
Root an. cond. $9.28646$
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 − 8·19-s − 11·31-s − 37-s + 13·43-s + 9·49-s + 14·61-s − 5·67-s + 17·73-s + 13·79-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 − 1.83·19-s − 1.97·31-s − 0.164·37-s + 1.98·43-s + 9/7·49-s + 1.79·61-s − 0.610·67-s + 1.98·73-s + 1.46·79-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

\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10800\)    =    \(2^{4} \cdot 3^{3} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(86.2384\)
Root analytic conductor: \(9.28646\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 10800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.684167645\)
\(L(\frac12)\) \(\approx\) \(2.684167645\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 11 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 13 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 17 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
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

−16.68748517653033, −15.85316720337139, −15.39305412086244, −14.67086403923793, −14.39251399098995, −13.77867444875837, −12.95263153273884, −12.68376889220364, −11.77327928671815, −11.14244899937137, −10.84899546378496, −10.40146070729823, −9.218734846605993, −8.840169117282596, −8.215196604777232, −7.748314453210964, −6.937031976132548, −6.181676976874526, −5.571747043050847, −4.854681129553304, −4.083361486722713, −3.634736159971652, −2.272431024303962, −1.801204706581691, −0.7848998715904747, 0.7848998715904747, 1.801204706581691, 2.272431024303962, 3.634736159971652, 4.083361486722713, 4.854681129553304, 5.571747043050847, 6.181676976874526, 6.937031976132548, 7.748314453210964, 8.215196604777232, 8.840169117282596, 9.218734846605993, 10.40146070729823, 10.84899546378496, 11.14244899937137, 11.77327928671815, 12.68376889220364, 12.95263153273884, 13.77867444875837, 14.39251399098995, 14.67086403923793, 15.39305412086244, 15.85316720337139, 16.68748517653033

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