# Properties

 Label 2-2352-7.4-c1-0-23 Degree $2$ Conductor $2352$ Sign $0.605 + 0.795i$ Analytic cond. $18.7808$ Root an. cond. $4.33368$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (0.5 − 0.866i)3-s + (1 + 1.73i)5-s + (−0.499 − 0.866i)9-s + (2 − 3.46i)11-s − 2·13-s + 1.99·15-s + (3 − 5.19i)17-s + (2 + 3.46i)19-s + (0.500 − 0.866i)25-s − 0.999·27-s − 2·29-s + (−1.99 − 3.46i)33-s + (−3 − 5.19i)37-s + (−1 + 1.73i)39-s + 2·41-s + ⋯
 L(s)  = 1 + (0.288 − 0.499i)3-s + (0.447 + 0.774i)5-s + (−0.166 − 0.288i)9-s + (0.603 − 1.04i)11-s − 0.554·13-s + 0.516·15-s + (0.727 − 1.26i)17-s + (0.458 + 0.794i)19-s + (0.100 − 0.173i)25-s − 0.192·27-s − 0.371·29-s + (−0.348 − 0.603i)33-s + (−0.493 − 0.854i)37-s + (−0.160 + 0.277i)39-s + 0.312·41-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2352$$    =    $$2^{4} \cdot 3 \cdot 7^{2}$$ Sign: $0.605 + 0.795i$ Analytic conductor: $$18.7808$$ Root analytic conductor: $$4.33368$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{2352} (1537, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 2352,\ (\ :1/2),\ 0.605 + 0.795i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.166858736$$ $$L(\frac12)$$ $$\approx$$ $$2.166858736$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 + (-0.5 + 0.866i)T$$
7 $$1$$
good5 $$1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2}$$
11 $$1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2}$$
13 $$1 + 2T + 13T^{2}$$
17 $$1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + (-11.5 + 19.9i)T^{2}$$
29 $$1 + 2T + 29T^{2}$$
31 $$1 + (-15.5 - 26.8i)T^{2}$$
37 $$1 + (3 + 5.19i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 - 2T + 41T^{2}$$
43 $$1 - 4T + 43T^{2}$$
47 $$1 + (-23.5 + 40.7i)T^{2}$$
53 $$1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2}$$
71 $$1 + 71T^{2}$$
73 $$1 + (-3 + 5.19i)T + (-36.5 - 63.2i)T^{2}$$
79 $$1 + (8 + 13.8i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 - 12T + 83T^{2}$$
89 $$1 + (-7 - 12.1i)T + (-44.5 + 77.0i)T^{2}$$
97 $$1 - 18T + 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

−8.986110837635124187554553854430, −7.912485992073285104694939895389, −7.39183450447668903519168005615, −6.54951996829193710377981229934, −5.90441797912309754295200006364, −5.07172018458693842342994873903, −3.69485898387662226937709894375, −3.02200297793579465275070352352, −2.10471627684087977317770597999, −0.78354883221851688463399060692, 1.25472338647344124511488483847, 2.24281320202841076040231901051, 3.45972651216623040377621903417, 4.36813759831279827187110938651, 5.04949071521281981685592426534, 5.79227885712163690276153584226, 6.85300057974494424073083374816, 7.60716961882873176513262596023, 8.525585438477936160443132809337, 9.141586222502462082609396679405