# Properties

 Label 2-390-5.4-c1-0-7 Degree $2$ Conductor $390$ Sign $-0.447 + 0.894i$ Analytic cond. $3.11416$ Root an. cond. $1.76469$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − i·2-s − i·3-s − 4-s + (2 + i)5-s − 6-s − 4i·7-s + i·8-s − 9-s + (1 − 2i)10-s + 2·11-s + i·12-s + i·13-s − 4·14-s + (1 − 2i)15-s + 16-s − 4i·17-s + ⋯
 L(s)  = 1 − 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (0.894 + 0.447i)5-s − 0.408·6-s − 1.51i·7-s + 0.353i·8-s − 0.333·9-s + (0.316 − 0.632i)10-s + 0.603·11-s + 0.288i·12-s + 0.277i·13-s − 1.06·14-s + (0.258 − 0.516i)15-s + 0.250·16-s − 0.970i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$390$$    =    $$2 \cdot 3 \cdot 5 \cdot 13$$ Sign: $-0.447 + 0.894i$ Analytic conductor: $$3.11416$$ Root analytic conductor: $$1.76469$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{390} (79, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 390,\ (\ :1/2),\ -0.447 + 0.894i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.763100 - 1.23472i$$ $$L(\frac12)$$ $$\approx$$ $$0.763100 - 1.23472i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + iT$$
3 $$1 + iT$$
5 $$1 + (-2 - i)T$$
13 $$1 - iT$$
good7 $$1 + 4iT - 7T^{2}$$
11 $$1 - 2T + 11T^{2}$$
17 $$1 + 4iT - 17T^{2}$$
19 $$1 + 2T + 19T^{2}$$
23 $$1 + 6iT - 23T^{2}$$
29 $$1 - 2T + 29T^{2}$$
31 $$1 + 4T + 31T^{2}$$
37 $$1 - 6iT - 37T^{2}$$
41 $$1 + 6T + 41T^{2}$$
43 $$1 - 8iT - 43T^{2}$$
47 $$1 + 8iT - 47T^{2}$$
53 $$1 - 10iT - 53T^{2}$$
59 $$1 - 14T + 59T^{2}$$
61 $$1 - 10T + 61T^{2}$$
67 $$1 + 4iT - 67T^{2}$$
71 $$1 - 8T + 71T^{2}$$
73 $$1 - 10iT - 73T^{2}$$
79 $$1 - 8T + 79T^{2}$$
83 $$1 - 12iT - 83T^{2}$$
89 $$1 - 18T + 89T^{2}$$
97 $$1 + 6iT - 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

−10.95030909764592178474522169338, −10.23845136228507693970389687210, −9.453993206689234124581425350749, −8.323560979338947872027091728120, −7.02555179057765223695230575436, −6.52567964516979694630917506526, −4.99600095108582412245908631971, −3.78286145293792027194094329732, −2.44101128541060251824225185472, −1.06042640914004630859042630322, 2.04586530366640465688845444023, 3.72794716259632351621374002454, 5.19510855767203085831627464049, 5.69097944702842112330053025984, 6.59800822985621947604709322250, 8.199621978548240832095296968422, 8.927157072855751300511983482411, 9.475364833362792739908843742432, 10.44950589932283994661408709602, 11.70541008195990028035299899017