Properties

Label 2-55545-1.1-c1-0-2
Degree $2$
Conductor $55545$
Sign $1$
Analytic cond. $443.529$
Root an. cond. $21.0601$
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  + 3-s − 2·4-s − 5-s + 7-s + 9-s − 4·11-s − 2·12-s + 3·13-s − 15-s + 4·16-s + 17-s + 2·20-s + 21-s + 25-s + 27-s − 2·28-s − 3·29-s − 6·31-s − 4·33-s − 35-s − 2·36-s + 6·37-s + 3·39-s + 12·41-s − 6·43-s + 8·44-s − 45-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.577·12-s + 0.832·13-s − 0.258·15-s + 16-s + 0.242·17-s + 0.447·20-s + 0.218·21-s + 1/5·25-s + 0.192·27-s − 0.377·28-s − 0.557·29-s − 1.07·31-s − 0.696·33-s − 0.169·35-s − 1/3·36-s + 0.986·37-s + 0.480·39-s + 1.87·41-s − 0.914·43-s + 1.20·44-s − 0.149·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55545 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55545 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55545\)    =    \(3 \cdot 5 \cdot 7 \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(443.529\)
Root analytic conductor: \(21.0601\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 55545,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.690841743\)
\(L(\frac12)\) \(\approx\) \(1.690841743\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 - T \)
23 \( 1 \)
good2 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 11 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−14.41002510537340, −13.73953394194163, −13.54697206442297, −12.86850684037288, −12.59932624469724, −12.00726510891917, −11.15447122506921, −10.74203920916458, −10.43091839156666, −9.418307359817429, −9.327523701848946, −8.706656061695909, −7.972188077746858, −7.836096751793204, −7.395397937646047, −6.386787293953621, −5.737274866663499, −5.265149991810291, −4.570783819620774, −4.123716380207762, −3.489887429282206, −2.946574395718199, −2.137687326361583, −1.271862350640893, −0.4717942214636832, 0.4717942214636832, 1.271862350640893, 2.137687326361583, 2.946574395718199, 3.489887429282206, 4.123716380207762, 4.570783819620774, 5.265149991810291, 5.737274866663499, 6.386787293953621, 7.395397937646047, 7.836096751793204, 7.972188077746858, 8.706656061695909, 9.327523701848946, 9.418307359817429, 10.43091839156666, 10.74203920916458, 11.15447122506921, 12.00726510891917, 12.59932624469724, 12.86850684037288, 13.54697206442297, 13.73953394194163, 14.41002510537340

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