# Properties

 Label 1-1148-1148.1147-r0-0-0 Degree $1$ Conductor $1148$ Sign $1$ Analytic cond. $5.33128$ Root an. cond. $5.33128$ Motivic weight $0$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 − 3-s − 5-s + 9-s + 11-s + 13-s + 15-s + 17-s − 19-s − 23-s + 25-s − 27-s − 29-s + 31-s − 33-s + 37-s − 39-s − 43-s − 45-s − 47-s − 51-s − 53-s − 55-s + 57-s + 59-s − 61-s − 65-s + 67-s + ⋯
 L(s)  = 1 − 3-s − 5-s + 9-s + 11-s + 13-s + 15-s + 17-s − 19-s − 23-s + 25-s − 27-s − 29-s + 31-s − 33-s + 37-s − 39-s − 43-s − 45-s − 47-s − 51-s − 53-s − 55-s + 57-s + 59-s − 61-s − 65-s + 67-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$1148$$    =    $$2^{2} \cdot 7 \cdot 41$$ Sign: $1$ Analytic conductor: $$5.33128$$ Root analytic conductor: $$5.33128$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: $\chi_{1148} (1147, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(1,\ 1148,\ (0:\ ),\ 1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.9132237921$$ $$L(\frac12)$$ $$\approx$$ $$0.9132237921$$ $$L(1)$$ $$\approx$$ $$0.7503779886$$ $$L(1)$$ $$\approx$$ $$0.7503779886$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
7 $$1$$
41 $$1$$
good3 $$1 - T$$
5 $$1 - T$$
11 $$1 + T$$
13 $$1 + T$$
17 $$1 + T$$
19 $$1 - T$$
23 $$1 - T$$
29 $$1 - T$$
31 $$1 + T$$
37 $$1 + T$$
43 $$1 - T$$
47 $$1 - T$$
53 $$1 - T$$
59 $$1 + T$$
61 $$1 - T$$
67 $$1 + T$$
71 $$1 + T$$
73 $$1 - T$$
79 $$1 + T$$
83 $$1 + T$$
89 $$1 + T$$
97 $$1 + T$$
show less
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−21.395400260215964572418489795782, −20.56540371450547115137984980887, −19.64249966480912472365623538878, −18.883344595689330269823569136256, −18.32118761330236334033417234263, −17.30148301429413037946829569061, −16.59698329293935501106756904521, −16.07125783923858819174720045805, −15.19142857651630923850528421799, −14.45947605168858943290584306041, −13.2836902220496356970819318459, −12.46548305823402100720057357517, −11.73948405025658552623162810310, −11.26551106254483217459168765751, −10.397540225264075492069959629797, −9.484880950440736484547650288944, −8.35790174887817223658843703418, −7.67289383837640347987866915967, −6.54685571688354696853427334719, −6.11605171693459882108355910101, −4.91762164290475360566942484658, −4.04998764575539115736630338004, −3.472940234776751711590552281902, −1.74461496277952618414248122792, −0.73433715700770444330418551446, 0.73433715700770444330418551446, 1.74461496277952618414248122792, 3.472940234776751711590552281902, 4.04998764575539115736630338004, 4.91762164290475360566942484658, 6.11605171693459882108355910101, 6.54685571688354696853427334719, 7.67289383837640347987866915967, 8.35790174887817223658843703418, 9.484880950440736484547650288944, 10.397540225264075492069959629797, 11.26551106254483217459168765751, 11.73948405025658552623162810310, 12.46548305823402100720057357517, 13.2836902220496356970819318459, 14.45947605168858943290584306041, 15.19142857651630923850528421799, 16.07125783923858819174720045805, 16.59698329293935501106756904521, 17.30148301429413037946829569061, 18.32118761330236334033417234263, 18.883344595689330269823569136256, 19.64249966480912472365623538878, 20.56540371450547115137984980887, 21.395400260215964572418489795782