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Properties

Label 1-153-153.70-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 $0$

Origins

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Normalization:  

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: $\chi_{153} (70, \cdot )$
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

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

Graph of the $Z$-function along the critical line