# Properties

 Label 2-4002-1.1-c1-0-57 Degree $2$ Conductor $4002$ Sign $-1$ Analytic cond. $31.9561$ Root an. cond. $5.65297$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s + 3-s + 4-s − 3.96·5-s − 6-s + 2.40·7-s − 8-s + 9-s + 3.96·10-s − 2.43·11-s + 12-s + 2.67·13-s − 2.40·14-s − 3.96·15-s + 16-s − 2.40·17-s − 18-s − 0.957·19-s − 3.96·20-s + 2.40·21-s + 2.43·22-s + 23-s − 24-s + 10.7·25-s − 2.67·26-s + 27-s + 2.40·28-s + ⋯
 L(s)  = 1 − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.77·5-s − 0.408·6-s + 0.908·7-s − 0.353·8-s + 0.333·9-s + 1.25·10-s − 0.733·11-s + 0.288·12-s + 0.743·13-s − 0.642·14-s − 1.02·15-s + 0.250·16-s − 0.582·17-s − 0.235·18-s − 0.219·19-s − 0.887·20-s + 0.524·21-s + 0.518·22-s + 0.208·23-s − 0.204·24-s + 2.14·25-s − 0.525·26-s + 0.192·27-s + 0.454·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$4002$$    =    $$2 \cdot 3 \cdot 23 \cdot 29$$ Sign: $-1$ Analytic conductor: $$31.9561$$ Root analytic conductor: $$5.65297$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 4002,\ (\ :1/2),\ -1)$$

## 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$$F_p(T)$
bad2 $$1 + T$$
3 $$1 - T$$
23 $$1 - T$$
29 $$1 + T$$
good5 $$1 + 3.96T + 5T^{2}$$
7 $$1 - 2.40T + 7T^{2}$$
11 $$1 + 2.43T + 11T^{2}$$
13 $$1 - 2.67T + 13T^{2}$$
17 $$1 + 2.40T + 17T^{2}$$
19 $$1 + 0.957T + 19T^{2}$$
31 $$1 + 7.81T + 31T^{2}$$
37 $$1 - 11.1T + 37T^{2}$$
41 $$1 - 4.38T + 41T^{2}$$
43 $$1 + 7.33T + 43T^{2}$$
47 $$1 + 2.08T + 47T^{2}$$
53 $$1 - 11.1T + 53T^{2}$$
59 $$1 + 5.54T + 59T^{2}$$
61 $$1 + 3.56T + 61T^{2}$$
67 $$1 - 13.5T + 67T^{2}$$
71 $$1 + 14.3T + 71T^{2}$$
73 $$1 + 15.4T + 73T^{2}$$
79 $$1 + 10.6T + 79T^{2}$$
83 $$1 + 6.81T + 83T^{2}$$
89 $$1 - 10.5T + 89T^{2}$$
97 $$1 - 14.5T + 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

−8.166852463966261437952439876979, −7.53213403956312607610880950516, −7.13556393320829806712384199471, −5.98654438809930217230344046299, −4.84165876630304447492375317761, −4.16732789139851539532989833934, −3.39207870158168823273105496037, −2.45659166964973097413540269864, −1.27863216648821612432368527292, 0, 1.27863216648821612432368527292, 2.45659166964973097413540269864, 3.39207870158168823273105496037, 4.16732789139851539532989833934, 4.84165876630304447492375317761, 5.98654438809930217230344046299, 7.13556393320829806712384199471, 7.53213403956312607610880950516, 8.166852463966261437952439876979