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Properties

Label 2-3627-403.402-c0-0-1
Degree $2$
Conductor $3627$
Sign $1$
Analytic cond. $1.81010$
Root an. cond. $1.34540$
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  + 4-s + 13-s + 16-s + 25-s − 31-s − 2·37-s + 49-s + 52-s + 64-s + 2·73-s + 100-s − 2·103-s + ⋯
L(s)  = 1  + 4-s + 13-s + 16-s + 25-s − 31-s − 2·37-s + 49-s + 52-s + 64-s + 2·73-s + 100-s − 2·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3627 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3627 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3627\)    =    \(3^{2} \cdot 13 \cdot 31\)
Sign: $1$
Analytic conductor: \(1.81010\)
Root analytic conductor: \(1.34540\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3627} (3223, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3627,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.747834690\)
\(L(\frac12)\) \(\approx\) \(1.747834690\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 - T \)
31 \( 1 + T \)
good2 \( ( 1 - T )( 1 + T ) \)
5 \( ( 1 - T )( 1 + T ) \)
7 \( ( 1 - T )( 1 + T ) \)
11 \( 1 + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 + T )^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )( 1 + T ) \)
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

−8.652780848713644008995791746949, −7.976268771300119578707015856030, −7.06930524813520474261456609102, −6.65762892363926092629995383948, −5.77313980255793759681157416402, −5.13246231122633836614835904404, −3.88106087403209776409078202112, −3.23020756453018667610569258898, −2.20687983029654535943934494562, −1.26208210814985655258980774138, 1.26208210814985655258980774138, 2.20687983029654535943934494562, 3.23020756453018667610569258898, 3.88106087403209776409078202112, 5.13246231122633836614835904404, 5.77313980255793759681157416402, 6.65762892363926092629995383948, 7.06930524813520474261456609102, 7.976268771300119578707015856030, 8.652780848713644008995791746949

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