# Properties

 Label 2-1617-1.1-c3-0-142 Degree $2$ Conductor $1617$ Sign $-1$ Analytic cond. $95.4060$ Root an. cond. $9.76760$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

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## Dirichlet series

 L(s)  = 1 − 1.36·2-s − 3·3-s − 6.14·4-s + 14.9·5-s + 4.09·6-s + 19.2·8-s + 9·9-s − 20.3·10-s + 11·11-s + 18.4·12-s − 36.0·13-s − 44.7·15-s + 22.8·16-s + 59.4·17-s − 12.2·18-s − 69.7·19-s − 91.6·20-s − 15.0·22-s − 29.1·23-s − 57.8·24-s + 97.8·25-s + 49.1·26-s − 27·27-s + 34.4·29-s + 61.0·30-s + 309.·31-s − 185.·32-s + ⋯
 L(s)  = 1 − 0.482·2-s − 0.577·3-s − 0.767·4-s + 1.33·5-s + 0.278·6-s + 0.852·8-s + 0.333·9-s − 0.643·10-s + 0.301·11-s + 0.443·12-s − 0.768·13-s − 0.770·15-s + 0.356·16-s + 0.848·17-s − 0.160·18-s − 0.841·19-s − 1.02·20-s − 0.145·22-s − 0.264·23-s − 0.492·24-s + 0.782·25-s + 0.370·26-s − 0.192·27-s + 0.220·29-s + 0.371·30-s + 1.79·31-s − 1.02·32-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1617$$    =    $$3 \cdot 7^{2} \cdot 11$$ Sign: $-1$ Analytic conductor: $$95.4060$$ Root analytic conductor: $$9.76760$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 1617,\ (\ :3/2),\ -1)$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + 3T$$
7 $$1$$
11 $$1 - 11T$$
good2 $$1 + 1.36T + 8T^{2}$$
5 $$1 - 14.9T + 125T^{2}$$
13 $$1 + 36.0T + 2.19e3T^{2}$$
17 $$1 - 59.4T + 4.91e3T^{2}$$
19 $$1 + 69.7T + 6.85e3T^{2}$$
23 $$1 + 29.1T + 1.21e4T^{2}$$
29 $$1 - 34.4T + 2.43e4T^{2}$$
31 $$1 - 309.T + 2.97e4T^{2}$$
37 $$1 + 347.T + 5.06e4T^{2}$$
41 $$1 + 407.T + 6.89e4T^{2}$$
43 $$1 + 168.T + 7.95e4T^{2}$$
47 $$1 + 568.T + 1.03e5T^{2}$$
53 $$1 + 146.T + 1.48e5T^{2}$$
59 $$1 - 458.T + 2.05e5T^{2}$$
61 $$1 - 638.T + 2.26e5T^{2}$$
67 $$1 + 179.T + 3.00e5T^{2}$$
71 $$1 + 31.5T + 3.57e5T^{2}$$
73 $$1 + 187.T + 3.89e5T^{2}$$
79 $$1 - 939.T + 4.93e5T^{2}$$
83 $$1 - 620.T + 5.71e5T^{2}$$
89 $$1 - 775.T + 7.04e5T^{2}$$
97 $$1 + 297.T + 9.12e5T^{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

−8.694316728726191429276449320805, −8.061970690671633742474670003992, −6.87842338781886373619408416053, −6.23695972690092341113581877109, −5.20767530487883282304478889045, −4.81663687012568181343121815271, −3.53946757281041759806558362573, −2.10759791365528759944463759770, −1.21107056740382640596734994596, 0, 1.21107056740382640596734994596, 2.10759791365528759944463759770, 3.53946757281041759806558362573, 4.81663687012568181343121815271, 5.20767530487883282304478889045, 6.23695972690092341113581877109, 6.87842338781886373619408416053, 8.061970690671633742474670003992, 8.694316728726191429276449320805