Properties

Label 2-7098-1.1-c1-0-130
Degree $2$
Conductor $7098$
Sign $-1$
Analytic cond. $56.6778$
Root an. cond. $7.52846$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s + 2.10·5-s − 6-s − 7-s − 8-s + 9-s − 2.10·10-s − 1.55·11-s + 12-s + 14-s + 2.10·15-s + 16-s − 1.24·17-s − 18-s − 6.98·19-s + 2.10·20-s − 21-s + 1.55·22-s − 1.97·23-s − 24-s − 0.553·25-s + 27-s − 28-s + 6.02·29-s − 2.10·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.943·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.666·10-s − 0.468·11-s + 0.288·12-s + 0.267·14-s + 0.544·15-s + 0.250·16-s − 0.302·17-s − 0.235·18-s − 1.60·19-s + 0.471·20-s − 0.218·21-s + 0.331·22-s − 0.411·23-s − 0.204·24-s − 0.110·25-s + 0.192·27-s − 0.188·28-s + 1.11·29-s − 0.384·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7098\)    =    \(2 \cdot 3 \cdot 7 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(56.6778\)
Root analytic conductor: \(7.52846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7098} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7098,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
7 \( 1 + T \)
13 \( 1 \)
good5 \( 1 - 2.10T + 5T^{2} \)
11 \( 1 + 1.55T + 11T^{2} \)
17 \( 1 + 1.24T + 17T^{2} \)
19 \( 1 + 6.98T + 19T^{2} \)
23 \( 1 + 1.97T + 23T^{2} \)
29 \( 1 - 6.02T + 29T^{2} \)
31 \( 1 + 2.75T + 31T^{2} \)
37 \( 1 - 2.76T + 37T^{2} \)
41 \( 1 - 7.97T + 41T^{2} \)
43 \( 1 + 3.66T + 43T^{2} \)
47 \( 1 + 5.63T + 47T^{2} \)
53 \( 1 - 8.81T + 53T^{2} \)
59 \( 1 + 5.40T + 59T^{2} \)
61 \( 1 - 3.42T + 61T^{2} \)
67 \( 1 - 1.40T + 67T^{2} \)
71 \( 1 + 11.6T + 71T^{2} \)
73 \( 1 - 7.48T + 73T^{2} \)
79 \( 1 + 16.4T + 79T^{2} \)
83 \( 1 + 16.7T + 83T^{2} \)
89 \( 1 + 10.9T + 89T^{2} \)
97 \( 1 - 6.39T + 97T^{2} \)
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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

−7.67982144116897673344273240688, −6.95602625737205252696877290751, −6.24080165792199094569599053991, −5.77379063266230486658802462736, −4.68127464828898868070376260144, −3.88370001566702896396144556471, −2.74837673376280236409416220452, −2.30260493663583147737363002950, −1.41863045474004995985773967114, 0, 1.41863045474004995985773967114, 2.30260493663583147737363002950, 2.74837673376280236409416220452, 3.88370001566702896396144556471, 4.68127464828898868070376260144, 5.77379063266230486658802462736, 6.24080165792199094569599053991, 6.95602625737205252696877290751, 7.67982144116897673344273240688

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