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Properties

Label 2-3520-55.54-c0-0-5
Degree $2$
Conductor $3520$
Sign $1$
Analytic cond. $1.75670$
Root an. cond. $1.32540$
Motivic weight $0$
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 + 9-s − 11-s + 25-s + 2·31-s + 45-s − 49-s − 55-s + 2·59-s − 2·71-s + 81-s − 2·89-s − 99-s + ⋯
L(s)  = 1  + 5-s + 9-s − 11-s + 25-s + 2·31-s + 45-s − 49-s − 55-s + 2·59-s − 2·71-s + 81-s − 2·89-s − 99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $1$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3520} (769, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3520,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.613417514\)
\(L(\frac12)\) \(\approx\) \(1.613417514\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
11 \( 1 + T \)
good3 \( ( 1 - T )( 1 + T ) \)
7 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )^{2} \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( 1 + T^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 + T )^{2} \)
73 \( 1 + T^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( 1 + T^{2} \)
89 \( ( 1 + T )^{2} \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−8.719266822540352489500311032610, −8.057758849817367764513826143291, −7.18769592256208494909565236663, −6.54022943116205175121072980883, −5.74619922537626995166740287438, −4.97093021531004683579711418447, −4.29664478890215556483001441402, −3.04559805417498339078981594820, −2.25726698986072320902260133286, −1.21245303565018344121967747062, 1.21245303565018344121967747062, 2.25726698986072320902260133286, 3.04559805417498339078981594820, 4.29664478890215556483001441402, 4.97093021531004683579711418447, 5.74619922537626995166740287438, 6.54022943116205175121072980883, 7.18769592256208494909565236663, 8.057758849817367764513826143291, 8.719266822540352489500311032610

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