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Properties

Label 2-3700-148.147-c0-0-6
Degree $2$
Conductor $3700$
Sign $1$
Analytic cond. $1.84654$
Root an. cond. $1.35887$
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  + 2-s + 4-s + 8-s + 9-s + 16-s + 18-s − 19-s − 23-s + 2·31-s + 32-s + 36-s + 37-s − 38-s − 41-s − 43-s − 46-s + 49-s − 53-s − 59-s + 2·62-s + 64-s + 72-s − 73-s + 74-s − 76-s − 79-s + 81-s + ⋯
L(s)  = 1  + 2-s + 4-s + 8-s + 9-s + 16-s + 18-s − 19-s − 23-s + 2·31-s + 32-s + 36-s + 37-s − 38-s − 41-s − 43-s − 46-s + 49-s − 53-s − 59-s + 2·62-s + 64-s + 72-s − 73-s + 74-s − 76-s − 79-s + 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3700\)    =    \(2^{2} \cdot 5^{2} \cdot 37\)
Sign: $1$
Analytic conductor: \(1.84654\)
Root analytic conductor: \(1.35887\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3700} (3551, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3700,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.691327877\)
\(L(\frac12)\) \(\approx\) \(2.691327877\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−8.438038724244311405770993811846, −7.88420378305325390075645747621, −6.99446324522723972533811963933, −6.42263979823660368993101691374, −5.75910621592356685767027135804, −4.59007463606938351211036862093, −4.36566813708242412163175731707, −3.33765426402307456970991998656, −2.36164885993655915262178801310, −1.42721421791403126229099306116, 1.42721421791403126229099306116, 2.36164885993655915262178801310, 3.33765426402307456970991998656, 4.36566813708242412163175731707, 4.59007463606938351211036862093, 5.75910621592356685767027135804, 6.42263979823660368993101691374, 6.99446324522723972533811963933, 7.88420378305325390075645747621, 8.438038724244311405770993811846

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