# Properties

 Label 2-115-1.1-c3-0-0 Degree $2$ Conductor $115$ Sign $1$ Analytic cond. $6.78521$ Root an. cond. $2.60484$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.93·2-s − 3.85·3-s + 0.637·4-s − 5·5-s + 11.3·6-s − 23.5·7-s + 21.6·8-s − 12.1·9-s + 14.6·10-s − 58.2·11-s − 2.45·12-s + 68.5·13-s + 69.3·14-s + 19.2·15-s − 68.6·16-s + 101.·17-s + 35.6·18-s − 7.02·19-s − 3.18·20-s + 91.0·21-s + 171.·22-s − 23·23-s − 83.4·24-s + 25·25-s − 201.·26-s + 150.·27-s − 15.0·28-s + ⋯
 L(s)  = 1 − 1.03·2-s − 0.742·3-s + 0.0797·4-s − 0.447·5-s + 0.771·6-s − 1.27·7-s + 0.956·8-s − 0.448·9-s + 0.464·10-s − 1.59·11-s − 0.0591·12-s + 1.46·13-s + 1.32·14-s + 0.332·15-s − 1.07·16-s + 1.44·17-s + 0.466·18-s − 0.0848·19-s − 0.0356·20-s + 0.945·21-s + 1.65·22-s − 0.208·23-s − 0.709·24-s + 0.200·25-s − 1.51·26-s + 1.07·27-s − 0.101·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$115$$    =    $$5 \cdot 23$$ Sign: $1$ Analytic conductor: $$6.78521$$ Root analytic conductor: $$2.60484$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 115,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.3393370062$$ $$L(\frac12)$$ $$\approx$$ $$0.3393370062$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1 + 5T$$
23 $$1 + 23T$$
good2 $$1 + 2.93T + 8T^{2}$$
3 $$1 + 3.85T + 27T^{2}$$
7 $$1 + 23.5T + 343T^{2}$$
11 $$1 + 58.2T + 1.33e3T^{2}$$
13 $$1 - 68.5T + 2.19e3T^{2}$$
17 $$1 - 101.T + 4.91e3T^{2}$$
19 $$1 + 7.02T + 6.85e3T^{2}$$
29 $$1 - 206.T + 2.43e4T^{2}$$
31 $$1 - 54.8T + 2.97e4T^{2}$$
37 $$1 + 241.T + 5.06e4T^{2}$$
41 $$1 + 122.T + 6.89e4T^{2}$$
43 $$1 + 320.T + 7.95e4T^{2}$$
47 $$1 - 107.T + 1.03e5T^{2}$$
53 $$1 - 127.T + 1.48e5T^{2}$$
59 $$1 - 693.T + 2.05e5T^{2}$$
61 $$1 + 899.T + 2.26e5T^{2}$$
67 $$1 - 110.T + 3.00e5T^{2}$$
71 $$1 - 225.T + 3.57e5T^{2}$$
73 $$1 - 746.T + 3.89e5T^{2}$$
79 $$1 + 1.09e3T + 4.93e5T^{2}$$
83 $$1 - 1.28e3T + 5.71e5T^{2}$$
89 $$1 - 1.20e3T + 7.04e5T^{2}$$
97 $$1 - 903.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

−12.99298136631232983892396359156, −11.88669253397764018252067523996, −10.57890989989705518926454502166, −10.16209450217894625077240365764, −8.715476569558740201865894845728, −7.890627998094180743156842988206, −6.45859579729192296268086441367, −5.22482506741171296884994348225, −3.29939336007389935685831012019, −0.59393817036756763483328392026, 0.59393817036756763483328392026, 3.29939336007389935685831012019, 5.22482506741171296884994348225, 6.45859579729192296268086441367, 7.890627998094180743156842988206, 8.715476569558740201865894845728, 10.16209450217894625077240365764, 10.57890989989705518926454502166, 11.88669253397764018252067523996, 12.99298136631232983892396359156