# Properties

 Label 2-2925-5.4-c1-0-41 Degree $2$ Conductor $2925$ Sign $-0.894 + 0.447i$ Analytic cond. $23.3562$ Root an. cond. $4.83282$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.56i·2-s − 4.56·4-s + 0.438i·7-s + 6.56i·8-s − 1.56·11-s − i·13-s + 1.12·14-s + 7.68·16-s − 1.56i·17-s + 5.12·19-s + 4i·22-s + 2.43i·23-s − 2.56·26-s − 2i·28-s + 7.12·29-s + ⋯
 L(s)  = 1 − 1.81i·2-s − 2.28·4-s + 0.165i·7-s + 2.31i·8-s − 0.470·11-s − 0.277i·13-s + 0.300·14-s + 1.92·16-s − 0.378i·17-s + 1.17·19-s + 0.852i·22-s + 0.508i·23-s − 0.502·26-s − 0.377i·28-s + 1.32·29-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2925$$    =    $$3^{2} \cdot 5^{2} \cdot 13$$ Sign: $-0.894 + 0.447i$ Analytic conductor: $$23.3562$$ Root analytic conductor: $$4.83282$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{2925} (2224, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 2925,\ (\ :1/2),\ -0.894 + 0.447i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.446602269$$ $$L(\frac12)$$ $$\approx$$ $$1.446602269$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
5 $$1$$
13 $$1 + iT$$
good2 $$1 + 2.56iT - 2T^{2}$$
7 $$1 - 0.438iT - 7T^{2}$$
11 $$1 + 1.56T + 11T^{2}$$
17 $$1 + 1.56iT - 17T^{2}$$
19 $$1 - 5.12T + 19T^{2}$$
23 $$1 - 2.43iT - 23T^{2}$$
29 $$1 - 7.12T + 29T^{2}$$
31 $$1 - 6T + 31T^{2}$$
37 $$1 - 10.6iT - 37T^{2}$$
41 $$1 - 3.56T + 41T^{2}$$
43 $$1 - 3.12iT - 43T^{2}$$
47 $$1 + 11.1iT - 47T^{2}$$
53 $$1 + 4.68iT - 53T^{2}$$
59 $$1 + 12T + 59T^{2}$$
61 $$1 + 6.68T + 61T^{2}$$
67 $$1 - 11.3iT - 67T^{2}$$
71 $$1 - 10.4T + 71T^{2}$$
73 $$1 + 6iT - 73T^{2}$$
79 $$1 + 4.68T + 79T^{2}$$
83 $$1 + 16.4iT - 83T^{2}$$
89 $$1 + 10.6T + 89T^{2}$$
97 $$1 + 16.9iT - 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

−8.636555408957354431999488313242, −8.059134490060930923849092268214, −7.03816708219163981007124875856, −5.83134494950060801168964073531, −4.96770664861745775233450572262, −4.38556998991923175014353566161, −3.12827078831167771823960325303, −2.89237118834506148333026905213, −1.66074262494975236278290587920, −0.66462675180913818179018144230, 0.874020332012693529786356045394, 2.69569270250612422945595442605, 3.95595790217648379341153352371, 4.68054411481477592986656529344, 5.41345288048321806826107657821, 6.18025790019049081302577031680, 6.75631365046335550018211651444, 7.71672022628864312031082754476, 7.901342193312102699757567088733, 8.939504912440241724357290625193