Properties

Label 2-1638-1.1-c3-0-74
Degree $2$
Conductor $1638$
Sign $-1$
Analytic cond. $96.6451$
Root an. cond. $9.83082$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 4·4-s + 9·5-s + 7·7-s − 8·8-s − 18·10-s + 18·11-s + 13·13-s − 14·14-s + 16·16-s − 60·17-s − 43·19-s + 36·20-s − 36·22-s − 9·23-s − 44·25-s − 26·26-s + 28·28-s + 249·29-s − 79·31-s − 32·32-s + 120·34-s + 63·35-s − 412·37-s + 86·38-s − 72·40-s − 222·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.804·5-s + 0.377·7-s − 0.353·8-s − 0.569·10-s + 0.493·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.856·17-s − 0.519·19-s + 0.402·20-s − 0.348·22-s − 0.0815·23-s − 0.351·25-s − 0.196·26-s + 0.188·28-s + 1.59·29-s − 0.457·31-s − 0.176·32-s + 0.605·34-s + 0.304·35-s − 1.83·37-s + 0.367·38-s − 0.284·40-s − 0.845·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-1$
Analytic conductor: \(96.6451\)
Root analytic conductor: \(9.83082\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1638} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1638,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p T \)
3 \( 1 \)
7 \( 1 - p T \)
13 \( 1 - p T \)
good5 \( 1 - 9 T + p^{3} T^{2} \)
11 \( 1 - 18 T + p^{3} T^{2} \)
17 \( 1 + 60 T + p^{3} T^{2} \)
19 \( 1 + 43 T + p^{3} T^{2} \)
23 \( 1 + 9 T + p^{3} T^{2} \)
29 \( 1 - 249 T + p^{3} T^{2} \)
31 \( 1 + 79 T + p^{3} T^{2} \)
37 \( 1 + 412 T + p^{3} T^{2} \)
41 \( 1 + 222 T + p^{3} T^{2} \)
43 \( 1 + 295 T + p^{3} T^{2} \)
47 \( 1 + 411 T + p^{3} T^{2} \)
53 \( 1 - 237 T + p^{3} T^{2} \)
59 \( 1 - 384 T + p^{3} T^{2} \)
61 \( 1 + 466 T + p^{3} T^{2} \)
67 \( 1 + 1042 T + p^{3} T^{2} \)
71 \( 1 - 288 T + p^{3} T^{2} \)
73 \( 1 + 691 T + p^{3} T^{2} \)
79 \( 1 - 1001 T + p^{3} T^{2} \)
83 \( 1 + 39 T + p^{3} T^{2} \)
89 \( 1 - 339 T + p^{3} T^{2} \)
97 \( 1 - 713 T + p^{3} T^{2} \)
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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

−8.697008695388896598516016265403, −8.069012336792659061747742028949, −6.86428762846836928770629558661, −6.45520218725832774252810656534, −5.45942598159436203494874749018, −4.50698625979662281459054887464, −3.30775496598806657471276755372, −2.09863302386225072051392185439, −1.43613961360454166384426843883, 0, 1.43613961360454166384426843883, 2.09863302386225072051392185439, 3.30775496598806657471276755372, 4.50698625979662281459054887464, 5.45942598159436203494874749018, 6.45520218725832774252810656534, 6.86428762846836928770629558661, 8.069012336792659061747742028949, 8.697008695388896598516016265403

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