Properties

Label 2-1400-1.1-c3-0-34
Degree $2$
Conductor $1400$
Sign $1$
Analytic cond. $82.6026$
Root an. cond. $9.08860$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·3-s + 7·7-s − 2·9-s + 11·11-s − 46·13-s + 127·17-s + 117·19-s + 35·21-s + 80·23-s − 145·27-s + 34·29-s − 292·31-s + 55·33-s − 376·37-s − 230·39-s + 507·41-s + 32·43-s − 134·47-s + 49·49-s + 635·51-s + 612·53-s + 585·57-s + 780·59-s − 426·61-s − 14·63-s − 207·67-s + 400·69-s + ⋯
L(s)  = 1  + 0.962·3-s + 0.377·7-s − 0.0740·9-s + 0.301·11-s − 0.981·13-s + 1.81·17-s + 1.41·19-s + 0.363·21-s + 0.725·23-s − 1.03·27-s + 0.217·29-s − 1.69·31-s + 0.290·33-s − 1.67·37-s − 0.944·39-s + 1.93·41-s + 0.113·43-s − 0.415·47-s + 1/7·49-s + 1.74·51-s + 1.58·53-s + 1.35·57-s + 1.72·59-s − 0.894·61-s − 0.0279·63-s − 0.377·67-s + 0.697·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(82.6026\)
Root analytic conductor: \(9.08860\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.351504385\)
\(L(\frac12)\) \(\approx\) \(3.351504385\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 - p T \)
good3 \( 1 - 5 T + p^{3} T^{2} \)
11 \( 1 - p T + p^{3} T^{2} \)
13 \( 1 + 46 T + p^{3} T^{2} \)
17 \( 1 - 127 T + p^{3} T^{2} \)
19 \( 1 - 117 T + p^{3} T^{2} \)
23 \( 1 - 80 T + p^{3} T^{2} \)
29 \( 1 - 34 T + p^{3} T^{2} \)
31 \( 1 + 292 T + p^{3} T^{2} \)
37 \( 1 + 376 T + p^{3} T^{2} \)
41 \( 1 - 507 T + p^{3} T^{2} \)
43 \( 1 - 32 T + p^{3} T^{2} \)
47 \( 1 + 134 T + p^{3} T^{2} \)
53 \( 1 - 612 T + p^{3} T^{2} \)
59 \( 1 - 780 T + p^{3} T^{2} \)
61 \( 1 + 426 T + p^{3} T^{2} \)
67 \( 1 + 207 T + p^{3} T^{2} \)
71 \( 1 - 702 T + p^{3} T^{2} \)
73 \( 1 - 1185 T + p^{3} T^{2} \)
79 \( 1 - 54 T + p^{3} T^{2} \)
83 \( 1 + 309 T + p^{3} T^{2} \)
89 \( 1 + 339 T + p^{3} T^{2} \)
97 \( 1 - 182 T + p^{3} T^{2} \)
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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

−9.261647182078908842298947724292, −8.382373405481993509497480074273, −7.54394743210727278564005689899, −7.18320087796204247247684281870, −5.62373960323670697667746967895, −5.16038649963684491806500135433, −3.75677558218323073922312354284, −3.11773722187425978147417132367, −2.09501309072694178817612454679, −0.888875161031208308347677340488, 0.888875161031208308347677340488, 2.09501309072694178817612454679, 3.11773722187425978147417132367, 3.75677558218323073922312354284, 5.16038649963684491806500135433, 5.62373960323670697667746967895, 7.18320087796204247247684281870, 7.54394743210727278564005689899, 8.382373405481993509497480074273, 9.261647182078908842298947724292

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