Properties

Label 2-7098-1.1-c1-0-111
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 − 1.44·5-s − 6-s + 7-s + 8-s + 9-s − 1.44·10-s − 3.74·11-s − 12-s + 14-s + 1.44·15-s + 16-s + 0.692·17-s + 18-s − 0.753·19-s − 1.44·20-s − 21-s − 3.74·22-s + 1.82·23-s − 24-s − 2.91·25-s − 27-s + 28-s + 6.89·29-s + 1.44·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.646·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.456·10-s − 1.12·11-s − 0.288·12-s + 0.267·14-s + 0.373·15-s + 0.250·16-s + 0.167·17-s + 0.235·18-s − 0.172·19-s − 0.323·20-s − 0.218·21-s − 0.797·22-s + 0.381·23-s − 0.204·24-s − 0.582·25-s − 0.192·27-s + 0.188·28-s + 1.28·29-s + 0.263·30-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 \( 1 + 1.44T + 5T^{2} \)
11 \( 1 + 3.74T + 11T^{2} \)
17 \( 1 - 0.692T + 17T^{2} \)
19 \( 1 + 0.753T + 19T^{2} \)
23 \( 1 - 1.82T + 23T^{2} \)
29 \( 1 - 6.89T + 29T^{2} \)
31 \( 1 - 0.960T + 31T^{2} \)
37 \( 1 + 3.04T + 37T^{2} \)
41 \( 1 - 1.44T + 41T^{2} \)
43 \( 1 - 5.82T + 43T^{2} \)
47 \( 1 + 3.71T + 47T^{2} \)
53 \( 1 + 1.57T + 53T^{2} \)
59 \( 1 - 6.65T + 59T^{2} \)
61 \( 1 + 7T + 61T^{2} \)
67 \( 1 + 9.11T + 67T^{2} \)
71 \( 1 - 0.219T + 71T^{2} \)
73 \( 1 + 4.91T + 73T^{2} \)
79 \( 1 + 7.44T + 79T^{2} \)
83 \( 1 + 8.75T + 83T^{2} \)
89 \( 1 + 14.5T + 89T^{2} \)
97 \( 1 + 6.06T + 97T^{2} \)
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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

−7.49488547361427080628330464525, −6.88614571631345116358310357259, −5.99837811627705000907029954961, −5.43589173504791265760953049906, −4.68508952431407788741268322472, −4.21076836877037232574123237924, −3.20584042931195142935833802741, −2.46090204217238709961918218358, −1.29354048399680351041744980482, 0, 1.29354048399680351041744980482, 2.46090204217238709961918218358, 3.20584042931195142935833802741, 4.21076836877037232574123237924, 4.68508952431407788741268322472, 5.43589173504791265760953049906, 5.99837811627705000907029954961, 6.88614571631345116358310357259, 7.49488547361427080628330464525

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