# Properties

 Label 2-63e2-441.103-c0-0-0 Degree $2$ Conductor $3969$ Sign $-0.620 + 0.784i$ Analytic cond. $1.98078$ Root an. cond. $1.40740$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (0.222 − 0.974i)4-s + (−0.365 + 0.930i)7-s + (−0.297 − 1.97i)13-s + (−0.900 − 0.433i)16-s + (−1.35 − 0.781i)19-s + (0.365 + 0.930i)25-s + (0.826 + 0.563i)28-s − 0.589i·31-s + (−1.82 − 0.563i)37-s + (−0.0546 − 0.728i)43-s + (−0.733 − 0.680i)49-s + (−1.98 − 0.149i)52-s + (1.32 − 0.302i)61-s + (−0.623 + 0.781i)64-s + 1.65·67-s + ⋯
 L(s)  = 1 + (0.222 − 0.974i)4-s + (−0.365 + 0.930i)7-s + (−0.297 − 1.97i)13-s + (−0.900 − 0.433i)16-s + (−1.35 − 0.781i)19-s + (0.365 + 0.930i)25-s + (0.826 + 0.563i)28-s − 0.589i·31-s + (−1.82 − 0.563i)37-s + (−0.0546 − 0.728i)43-s + (−0.733 − 0.680i)49-s + (−1.98 − 0.149i)52-s + (1.32 − 0.302i)61-s + (−0.623 + 0.781i)64-s + 1.65·67-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.620 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.620 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3969$$    =    $$3^{4} \cdot 7^{2}$$ Sign: $-0.620 + 0.784i$ Analytic conductor: $$1.98078$$ Root analytic conductor: $$1.40740$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{3969} (838, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3969,\ (\ :0),\ -0.620 + 0.784i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.8491339512$$ $$L(\frac12)$$ $$\approx$$ $$0.8491339512$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
7 $$1 + (0.365 - 0.930i)T$$
good2 $$1 + (-0.222 + 0.974i)T^{2}$$
5 $$1 + (-0.365 - 0.930i)T^{2}$$
11 $$1 + (0.955 - 0.294i)T^{2}$$
13 $$1 + (0.297 + 1.97i)T + (-0.955 + 0.294i)T^{2}$$
17 $$1 + (-0.0747 + 0.997i)T^{2}$$
19 $$1 + (1.35 + 0.781i)T + (0.5 + 0.866i)T^{2}$$
23 $$1 + (0.826 - 0.563i)T^{2}$$
29 $$1 + (0.0747 - 0.997i)T^{2}$$
31 $$1 + 0.589iT - T^{2}$$
37 $$1 + (1.82 + 0.563i)T + (0.826 + 0.563i)T^{2}$$
41 $$1 + (0.988 + 0.149i)T^{2}$$
43 $$1 + (0.0546 + 0.728i)T + (-0.988 + 0.149i)T^{2}$$
47 $$1 + (0.222 - 0.974i)T^{2}$$
53 $$1 + (0.826 - 0.563i)T^{2}$$
59 $$1 + (-0.623 - 0.781i)T^{2}$$
61 $$1 + (-1.32 + 0.302i)T + (0.900 - 0.433i)T^{2}$$
67 $$1 - 1.65T + T^{2}$$
71 $$1 + (-0.900 - 0.433i)T^{2}$$
73 $$1 + (-0.290 + 1.92i)T + (-0.955 - 0.294i)T^{2}$$
79 $$1 + 1.46T + T^{2}$$
83 $$1 + (-0.955 - 0.294i)T^{2}$$
89 $$1 + (0.733 - 0.680i)T^{2}$$
97 $$1 + (-1.61 + 0.930i)T + (0.5 - 0.866i)T^{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

−8.589926215392064471476238545685, −7.58598013487366822938313981210, −6.77967525891873255778617503069, −6.07582154753323648634790507106, −5.34606834896637775170143360208, −5.00149255329485039473688346735, −3.61611358563654612579927853971, −2.68941510415299302864919279222, −1.95297044228519951975089663064, −0.43963459911706736213690792734, 1.67081107853750593970545667347, 2.56739494023270630183016171024, 3.73563150176303343266960992736, 4.12250454981637963034245387967, 4.89633241778651316173784730531, 6.35270847419347799482863232861, 6.77475415199006943463268467181, 7.24580487534744868439195035015, 8.327427226003140634323336795003, 8.628803212577953689267501501418