Properties

Label 2-87120-1.1-c1-0-1
Degree $2$
Conductor $87120$
Sign $1$
Analytic cond. $695.656$
Root an. cond. $26.3753$
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  − 5-s + 3·7-s − 8·17-s − 8·19-s + 25-s − 2·29-s − 6·31-s − 3·35-s − 5·41-s − 43-s − 5·47-s + 2·49-s − 8·53-s − 10·59-s + 7·61-s + 7·67-s − 14·71-s − 16·73-s − 10·79-s − 12·83-s + 8·85-s − 9·89-s + 8·95-s + 12·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.13·7-s − 1.94·17-s − 1.83·19-s + 1/5·25-s − 0.371·29-s − 1.07·31-s − 0.507·35-s − 0.780·41-s − 0.152·43-s − 0.729·47-s + 2/7·49-s − 1.09·53-s − 1.30·59-s + 0.896·61-s + 0.855·67-s − 1.66·71-s − 1.87·73-s − 1.12·79-s − 1.31·83-s + 0.867·85-s − 0.953·89-s + 0.820·95-s + 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(87120\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(695.656\)
Root analytic conductor: \(26.3753\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{87120} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 87120,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2964123226\)
\(L(\frac12)\) \(\approx\) \(0.2964123226\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
11 \( 1 \)
good7 \( 1 - 3 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 5 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−13.98916461507690, −13.27858887136401, −12.87155746112380, −12.62909533170985, −11.66138994900278, −11.41748295128410, −11.05040229131150, −10.55901083474553, −10.06425273680570, −9.179837831635889, −8.778667927498742, −8.432875599277081, −7.940950335541198, −7.259157664891349, −6.827503032974617, −6.264748897028920, −5.655412953821970, −4.893339914281054, −4.404075520872665, −4.215415785806540, −3.332420302769349, −2.568878013957965, −1.831753732043101, −1.577510861414626, −0.1628418465714272, 0.1628418465714272, 1.577510861414626, 1.831753732043101, 2.568878013957965, 3.332420302769349, 4.215415785806540, 4.404075520872665, 4.893339914281054, 5.655412953821970, 6.264748897028920, 6.827503032974617, 7.259157664891349, 7.940950335541198, 8.432875599277081, 8.778667927498742, 9.179837831635889, 10.06425273680570, 10.55901083474553, 11.05040229131150, 11.41748295128410, 11.66138994900278, 12.62909533170985, 12.87155746112380, 13.27858887136401, 13.98916461507690

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