Properties

Label 2-7e2-1.1-c5-0-12
Degree $2$
Conductor $49$
Sign $-1$
Analytic cond. $7.85880$
Root an. cond. $2.80335$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.81·2-s + 6.54·3-s − 24.0·4-s − 45.9·5-s + 18.4·6-s − 157.·8-s − 200.·9-s − 129.·10-s − 551.·11-s − 157.·12-s + 1.09e3·13-s − 301.·15-s + 326.·16-s − 1.18e3·17-s − 563.·18-s − 1.16e3·19-s + 1.10e3·20-s − 1.55e3·22-s + 44.3·23-s − 1.03e3·24-s − 1.00e3·25-s + 3.07e3·26-s − 2.90e3·27-s + 3.32e3·29-s − 847.·30-s + 8.78e3·31-s + 5.96e3·32-s + ⋯
L(s)  = 1  + 0.497·2-s + 0.420·3-s − 0.752·4-s − 0.822·5-s + 0.209·6-s − 0.872·8-s − 0.823·9-s − 0.409·10-s − 1.37·11-s − 0.316·12-s + 1.79·13-s − 0.345·15-s + 0.318·16-s − 0.990·17-s − 0.409·18-s − 0.741·19-s + 0.618·20-s − 0.684·22-s + 0.0174·23-s − 0.366·24-s − 0.323·25-s + 0.893·26-s − 0.765·27-s + 0.735·29-s − 0.171·30-s + 1.64·31-s + 1.03·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $-1$
Analytic conductor: \(7.85880\)
Root analytic conductor: \(2.80335\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 49,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
good2 \( 1 - 2.81T + 32T^{2} \)
3 \( 1 - 6.54T + 243T^{2} \)
5 \( 1 + 45.9T + 3.12e3T^{2} \)
11 \( 1 + 551.T + 1.61e5T^{2} \)
13 \( 1 - 1.09e3T + 3.71e5T^{2} \)
17 \( 1 + 1.18e3T + 1.41e6T^{2} \)
19 \( 1 + 1.16e3T + 2.47e6T^{2} \)
23 \( 1 - 44.3T + 6.43e6T^{2} \)
29 \( 1 - 3.32e3T + 2.05e7T^{2} \)
31 \( 1 - 8.78e3T + 2.86e7T^{2} \)
37 \( 1 + 2.55e3T + 6.93e7T^{2} \)
41 \( 1 + 1.27e4T + 1.15e8T^{2} \)
43 \( 1 + 96.7T + 1.47e8T^{2} \)
47 \( 1 - 7.67e3T + 2.29e8T^{2} \)
53 \( 1 + 1.19e4T + 4.18e8T^{2} \)
59 \( 1 - 9.85e3T + 7.14e8T^{2} \)
61 \( 1 + 3.85e4T + 8.44e8T^{2} \)
67 \( 1 + 6.75e4T + 1.35e9T^{2} \)
71 \( 1 + 6.13e4T + 1.80e9T^{2} \)
73 \( 1 - 1.85e3T + 2.07e9T^{2} \)
79 \( 1 + 8.52T + 3.07e9T^{2} \)
83 \( 1 + 9.50e4T + 3.93e9T^{2} \)
89 \( 1 - 5.36e4T + 5.58e9T^{2} \)
97 \( 1 - 3.11e3T + 8.58e9T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.76339493296830452763038246471, −13.24634674106337374779345853472, −11.81213418936736043791719041296, −10.58241922657192764114216786963, −8.717782316119673337279916098100, −8.183056734645235509753885554998, −6.03373819024320781084429844665, −4.44770378485442115297806229410, −3.09666836325306029808568338199, 0, 3.09666836325306029808568338199, 4.44770378485442115297806229410, 6.03373819024320781084429844665, 8.183056734645235509753885554998, 8.717782316119673337279916098100, 10.58241922657192764114216786963, 11.81213418936736043791719041296, 13.24634674106337374779345853472, 13.76339493296830452763038246471

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