# Properties

 Label 2-6930-1.1-c1-0-27 Degree $2$ Conductor $6930$ Sign $1$ Analytic cond. $55.3363$ Root an. cond. $7.43883$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s − 11-s + 5.46·13-s − 14-s + 16-s − 3.46·17-s + 3.26·19-s + 20-s + 22-s − 2.19·23-s + 25-s − 5.46·26-s + 28-s + 1.26·29-s + 2·31-s − 32-s + 3.46·34-s + 35-s − 2.73·37-s − 3.26·38-s − 40-s − 8.19·41-s + ⋯
 L(s)  = 1 − 0.707·2-s + 0.5·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s − 0.301·11-s + 1.51·13-s − 0.267·14-s + 0.250·16-s − 0.840·17-s + 0.749·19-s + 0.223·20-s + 0.213·22-s − 0.457·23-s + 0.200·25-s − 1.07·26-s + 0.188·28-s + 0.235·29-s + 0.359·31-s − 0.176·32-s + 0.594·34-s + 0.169·35-s − 0.449·37-s − 0.530·38-s − 0.158·40-s − 1.28·41-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$6930$$    =    $$2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11$$ Sign: $1$ Analytic conductor: $$55.3363$$ Root analytic conductor: $$7.43883$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{6930} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 6930,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.762456610$$ $$L(\frac12)$$ $$\approx$$ $$1.762456610$$ $$L(\frac{3}{2})$$ 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$$
5 $$1 - T$$
7 $$1 - T$$
11 $$1 + T$$
good13 $$1 - 5.46T + 13T^{2}$$
17 $$1 + 3.46T + 17T^{2}$$
19 $$1 - 3.26T + 19T^{2}$$
23 $$1 + 2.19T + 23T^{2}$$
29 $$1 - 1.26T + 29T^{2}$$
31 $$1 - 2T + 31T^{2}$$
37 $$1 + 2.73T + 37T^{2}$$
41 $$1 + 8.19T + 41T^{2}$$
43 $$1 - 2T + 43T^{2}$$
47 $$1 - 6.92T + 47T^{2}$$
53 $$1 - 10.7T + 53T^{2}$$
59 $$1 - 6.92T + 59T^{2}$$
61 $$1 - 8.92T + 61T^{2}$$
67 $$1 + 4T + 67T^{2}$$
71 $$1 + 2.53T + 71T^{2}$$
73 $$1 - 6.39T + 73T^{2}$$
79 $$1 + 1.80T + 79T^{2}$$
83 $$1 + 4.39T + 83T^{2}$$
89 $$1 - 3.46T + 89T^{2}$$
97 $$1 + 16.5T + 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

−8.251938760688604696883783565364, −7.21839684304453786256179730672, −6.71119733089994755466345633731, −5.86688225846272254187912827572, −5.36326074032533073149393101568, −4.30578912311957278223543666010, −3.49208357857907319786904778050, −2.51212391152536451503767539070, −1.69057229522510183493212845516, −0.78387952088593746868922185064, 0.78387952088593746868922185064, 1.69057229522510183493212845516, 2.51212391152536451503767539070, 3.49208357857907319786904778050, 4.30578912311957278223543666010, 5.36326074032533073149393101568, 5.86688225846272254187912827572, 6.71119733089994755466345633731, 7.21839684304453786256179730672, 8.251938760688604696883783565364