# Properties

 Label 1-153-153.94-r0-0-0 Degree $1$ Conductor $153$ Sign $0.0407 + 0.999i$ Analytic cond. $0.710529$ Root an. cond. $0.710529$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + (−0.258 + 0.965i)5-s + (−0.258 − 0.965i)7-s + i·8-s + (−0.707 + 0.707i)10-s + (0.258 + 0.965i)11-s + (0.5 + 0.866i)13-s + (0.258 − 0.965i)14-s + (−0.5 + 0.866i)16-s − i·19-s + (−0.965 + 0.258i)20-s + (−0.258 + 0.965i)22-s + (−0.965 − 0.258i)23-s + (−0.866 − 0.5i)25-s + i·26-s + ⋯
 L(s)  = 1 + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + (−0.258 + 0.965i)5-s + (−0.258 − 0.965i)7-s + i·8-s + (−0.707 + 0.707i)10-s + (0.258 + 0.965i)11-s + (0.5 + 0.866i)13-s + (0.258 − 0.965i)14-s + (−0.5 + 0.866i)16-s − i·19-s + (−0.965 + 0.258i)20-s + (−0.258 + 0.965i)22-s + (−0.965 − 0.258i)23-s + (−0.866 − 0.5i)25-s + i·26-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0407 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0407 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$153$$    =    $$3^{2} \cdot 17$$ Sign: $0.0407 + 0.999i$ Analytic conductor: $$0.710529$$ Root analytic conductor: $$0.710529$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{153} (94, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 153,\ (0:\ ),\ 0.0407 + 0.999i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.206367435 + 1.158199078i$$ $$L(\frac12)$$ $$\approx$$ $$1.206367435 + 1.158199078i$$ $$L(1)$$ $$\approx$$ $$1.362414686 + 0.7368763967i$$ $$L(1)$$ $$\approx$$ $$1.362414686 + 0.7368763967i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
17 $$1$$
good2 $$1 + (0.866 + 0.5i)T$$
5 $$1 + (-0.258 + 0.965i)T$$
7 $$1 + (-0.258 - 0.965i)T$$
11 $$1 + (0.258 + 0.965i)T$$
13 $$1 + (0.5 + 0.866i)T$$
19 $$1 - iT$$
23 $$1 + (-0.965 - 0.258i)T$$
29 $$1 + (0.965 - 0.258i)T$$
31 $$1 + (0.258 - 0.965i)T$$
37 $$1 + (0.707 + 0.707i)T$$
41 $$1 + (0.965 + 0.258i)T$$
43 $$1 + (-0.866 - 0.5i)T$$
47 $$1 + (0.5 - 0.866i)T$$
53 $$1 - iT$$
59 $$1 + (0.866 - 0.5i)T$$
61 $$1 + (-0.258 - 0.965i)T$$
67 $$1 + (-0.5 - 0.866i)T$$
71 $$1 + (0.707 + 0.707i)T$$
73 $$1 + (-0.707 - 0.707i)T$$
79 $$1 + (0.258 + 0.965i)T$$
83 $$1 + (0.866 + 0.5i)T$$
89 $$1 - T$$
97 $$1 + (0.965 - 0.258i)T$$
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$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$

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

−28.01582116967432153991072574623, −27.186673973838139136198863746610, −25.27688158590529824961856767079, −24.78930569699577781765934144129, −23.77635566986236462732275825069, −22.833142515013807886716592648029, −21.74635438078852741162623418813, −21.041253239237509200462937087809, −19.961134050563004166506867881398, −19.186225634890112216092864506445, −18.01285959934843169190471162892, −16.24605625094344565680219222026, −15.80753365319148961223711186409, −14.50780853194626530052824223416, −13.378475687343655540471041803933, −12.42794149178951671133138745373, −11.768534326933818649530748124235, −10.46521144752427094434275056524, −9.14998836954815073166275213005, −8.08078524133850262112135156568, −6.09047768609382550300610567463, −5.48312281389038818116960420292, −4.071696806763438139579651654666, −2.9069581731389716363773043261, −1.25153371554648601017404551353, 2.32495062425189893510341452636, 3.745166766300942638229131423399, 4.5410424592011586477878248607, 6.40531290680219406447877715831, 6.94037114362795781919355484464, 8.029368297023954176897560890156, 9.81128082026742737627550992605, 11.06386525431447401346019739391, 11.96271832372575646612384716927, 13.34897709318028407049848117823, 14.10367750043611160292885952950, 15.061939819751728128365234532520, 16.01542773885205166880680083256, 17.111262902054462296430242955792, 18.1025277191271208085936173623, 19.56347403267101857295889602900, 20.43189547006438245657724260012, 21.67078695482391373840501613775, 22.55013709742237488666853300578, 23.32040249539577268883359659571, 24.00953723471003920733976571900, 25.471730909102230172720005020252, 26.14254137702780792913685093332, 26.82535537183694745572984838030, 28.341037705971082722251238095751