# Properties

 Label 1-107-107.106-r1-0-0 Degree $1$ Conductor $107$ Sign $1$ Analytic cond. $11.4987$ Root an. cond. $11.4987$ Motivic weight $0$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 − 2-s + 3-s + 4-s − 5-s − 6-s − 7-s − 8-s + 9-s + 10-s + 11-s + 12-s + 13-s + 14-s − 15-s + 16-s − 17-s − 18-s + 19-s − 20-s − 21-s − 22-s + 23-s − 24-s + 25-s − 26-s + 27-s − 28-s + ⋯
 L(s)  = 1 − 2-s + 3-s + 4-s − 5-s − 6-s − 7-s − 8-s + 9-s + 10-s + 11-s + 12-s + 13-s + 14-s − 15-s + 16-s − 17-s − 18-s + 19-s − 20-s − 21-s − 22-s + 23-s − 24-s + 25-s − 26-s + 27-s − 28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$107$$ Sign: $1$ Analytic conductor: $$11.4987$$ Root analytic conductor: $$11.4987$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: $\chi_{107} (106, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(1,\ 107,\ (1:\ ),\ 1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.326852223$$ $$L(\frac12)$$ $$\approx$$ $$1.326852223$$ $$L(1)$$ $$\approx$$ $$0.9111276755$$ $$L(1)$$ $$\approx$$ $$0.9111276755$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad107 $$1$$
good2 $$1 - T$$
3 $$1 + T$$
5 $$1 - T$$
7 $$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$$
41 $$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

−29.2507140111264096034499664478, −28.14960124760015303204373695397, −27.04871897087953847297680943008, −26.49471452647039048371369461478, −25.39703258965952156230327048379, −24.680981528594938389175505521, −23.39650356421593954043645065110, −21.96844673492842249095618647735, −20.44994622216287872583535436397, −19.81318600232194171014385393698, −19.12980309394917793974016224391, −18.148053187499111510642509674952, −16.44276709701150967191770631156, −15.77499870531388299215492066204, −14.82590541909745539391871807166, −13.26211254951704896426933380367, −11.97180264986610149206081297316, −10.75367821674543183729850651015, −9.304695466462708620345463212961, −8.72812555651843937650658447358, −7.41713536583272743899328295646, −6.52750358073905121597717020075, −3.9027623586238545483928887445, −2.90812216403723678026329739840, −1.01532045680624064311140232848, 1.01532045680624064311140232848, 2.90812216403723678026329739840, 3.9027623586238545483928887445, 6.52750358073905121597717020075, 7.41713536583272743899328295646, 8.72812555651843937650658447358, 9.304695466462708620345463212961, 10.75367821674543183729850651015, 11.97180264986610149206081297316, 13.26211254951704896426933380367, 14.82590541909745539391871807166, 15.77499870531388299215492066204, 16.44276709701150967191770631156, 18.148053187499111510642509674952, 19.12980309394917793974016224391, 19.81318600232194171014385393698, 20.44994622216287872583535436397, 21.96844673492842249095618647735, 23.39650356421593954043645065110, 24.680981528594938389175505521, 25.39703258965952156230327048379, 26.49471452647039048371369461478, 27.04871897087953847297680943008, 28.14960124760015303204373695397, 29.2507140111264096034499664478