# Properties

 Label 2-1875-1.1-c1-0-48 Degree $2$ Conductor $1875$ Sign $-1$ Analytic cond. $14.9719$ Root an. cond. $3.86936$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

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## Dirichlet series

 L(s)  = 1 − 2.68·2-s − 3-s + 5.22·4-s + 2.68·6-s + 1.68·7-s − 8.65·8-s + 9-s + 1.07·11-s − 5.22·12-s + 2.67·13-s − 4.53·14-s + 12.8·16-s − 3.93·17-s − 2.68·18-s − 1.17·19-s − 1.68·21-s − 2.89·22-s − 4.06·23-s + 8.65·24-s − 7.19·26-s − 27-s + 8.80·28-s + 5.95·29-s − 7.10·31-s − 17.1·32-s − 1.07·33-s + 10.5·34-s + ⋯
 L(s)  = 1 − 1.90·2-s − 0.577·3-s + 2.61·4-s + 1.09·6-s + 0.637·7-s − 3.05·8-s + 0.333·9-s + 0.325·11-s − 1.50·12-s + 0.742·13-s − 1.21·14-s + 3.20·16-s − 0.953·17-s − 0.633·18-s − 0.270·19-s − 0.368·21-s − 0.618·22-s − 0.847·23-s + 1.76·24-s − 1.41·26-s − 0.192·27-s + 1.66·28-s + 1.10·29-s − 1.27·31-s − 3.02·32-s − 0.187·33-s + 1.81·34-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1875$$    =    $$3 \cdot 5^{4}$$ Sign: $-1$ Analytic conductor: $$14.9719$$ Root analytic conductor: $$3.86936$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 1875,\ (\ :1/2),\ -1)$$

## Particular Values

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

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + T$$
5 $$1$$
good2 $$1 + 2.68T + 2T^{2}$$
7 $$1 - 1.68T + 7T^{2}$$
11 $$1 - 1.07T + 11T^{2}$$
13 $$1 - 2.67T + 13T^{2}$$
17 $$1 + 3.93T + 17T^{2}$$
19 $$1 + 1.17T + 19T^{2}$$
23 $$1 + 4.06T + 23T^{2}$$
29 $$1 - 5.95T + 29T^{2}$$
31 $$1 + 7.10T + 31T^{2}$$
37 $$1 + 4.58T + 37T^{2}$$
41 $$1 + 11.2T + 41T^{2}$$
43 $$1 - 2.58T + 43T^{2}$$
47 $$1 + 1.91T + 47T^{2}$$
53 $$1 + 2.54T + 53T^{2}$$
59 $$1 - 1.33T + 59T^{2}$$
61 $$1 + 7.28T + 61T^{2}$$
67 $$1 - 12.4T + 67T^{2}$$
71 $$1 + 5.98T + 71T^{2}$$
73 $$1 + 3.30T + 73T^{2}$$
79 $$1 + 4.00T + 79T^{2}$$
83 $$1 - 8.87T + 83T^{2}$$
89 $$1 - 15.4T + 89T^{2}$$
97 $$1 - 10.7T + 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

−8.748298348507309658513881274966, −8.299918635788798208327651840634, −7.43560564742744717456074152400, −6.62364071826257437210144552415, −6.11336251240664284501718849224, −4.92097941843277901803372304721, −3.58551772316829954831444845551, −2.13483728993346478922184714353, −1.36883888584412572952703562600, 0, 1.36883888584412572952703562600, 2.13483728993346478922184714353, 3.58551772316829954831444845551, 4.92097941843277901803372304721, 6.11336251240664284501718849224, 6.62364071826257437210144552415, 7.43560564742744717456074152400, 8.299918635788798208327651840634, 8.748298348507309658513881274966