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Properties

Label 1-161-161.16-r0-0-0
Degree $1$
Conductor $161$
Sign $0.00399 + 0.999i$
Analytic cond. $0.747680$
Root an. cond. $0.747680$
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  + (−0.786 + 0.618i)2-s + (0.580 + 0.814i)3-s + (0.235 − 0.971i)4-s + (0.981 + 0.189i)5-s + (−0.959 − 0.281i)6-s + (0.415 + 0.909i)8-s + (−0.327 + 0.945i)9-s + (−0.888 + 0.458i)10-s + (−0.786 − 0.618i)11-s + (0.928 − 0.371i)12-s + (0.841 + 0.540i)13-s + (0.415 + 0.909i)15-s + (−0.888 − 0.458i)16-s + (0.723 − 0.690i)17-s + (−0.327 − 0.945i)18-s + (0.723 + 0.690i)19-s + ⋯
L(s)  = 1  + (−0.786 + 0.618i)2-s + (0.580 + 0.814i)3-s + (0.235 − 0.971i)4-s + (0.981 + 0.189i)5-s + (−0.959 − 0.281i)6-s + (0.415 + 0.909i)8-s + (−0.327 + 0.945i)9-s + (−0.888 + 0.458i)10-s + (−0.786 − 0.618i)11-s + (0.928 − 0.371i)12-s + (0.841 + 0.540i)13-s + (0.415 + 0.909i)15-s + (−0.888 − 0.458i)16-s + (0.723 − 0.690i)17-s + (−0.327 − 0.945i)18-s + (0.723 + 0.690i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00399 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00399 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.00399 + 0.999i$
Analytic conductor: \(0.747680\)
Root analytic conductor: \(0.747680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (0:\ ),\ 0.00399 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7552634399 + 0.7582896992i\)
\(L(\frac12)\) \(\approx\) \(0.7552634399 + 0.7582896992i\)
\(L(1)\) \(\approx\) \(0.8631058594 + 0.5147474843i\)
\(L(1)\) \(\approx\) \(0.8631058594 + 0.5147474843i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.786 + 0.618i)T \)
3 \( 1 + (0.580 + 0.814i)T \)
5 \( 1 + (0.981 + 0.189i)T \)
11 \( 1 + (-0.786 - 0.618i)T \)
13 \( 1 + (0.841 + 0.540i)T \)
17 \( 1 + (0.723 - 0.690i)T \)
19 \( 1 + (0.723 + 0.690i)T \)
29 \( 1 + (-0.959 - 0.281i)T \)
31 \( 1 + (-0.995 - 0.0950i)T \)
37 \( 1 + (-0.327 + 0.945i)T \)
41 \( 1 + (-0.654 + 0.755i)T \)
43 \( 1 + (0.415 - 0.909i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (0.0475 + 0.998i)T \)
59 \( 1 + (-0.888 + 0.458i)T \)
61 \( 1 + (0.580 - 0.814i)T \)
67 \( 1 + (0.928 + 0.371i)T \)
71 \( 1 + (-0.142 - 0.989i)T \)
73 \( 1 + (0.235 - 0.971i)T \)
79 \( 1 + (0.0475 - 0.998i)T \)
83 \( 1 + (-0.654 - 0.755i)T \)
89 \( 1 + (-0.995 + 0.0950i)T \)
97 \( 1 + (-0.654 + 0.755i)T \)
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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

−27.799315105048843726412272430699, −26.14083823588149712857353020558, −25.87206101462358835710771647513, −24.99540375800611165130609156574, −23.90701157321036009858519973836, −22.5374503932844601023139987484, −21.187288888220709959889319839760, −20.60566471610331662599483427950, −19.71484008627559323275024682657, −18.44647688633748363420988020883, −18.01617837689976055808420830487, −17.08397613387127471111966937241, −15.70937918257796181140472939231, −14.24943610988245214033461488689, −12.98352158617524767834523730049, −12.72759787528255484389158626277, −11.1346967774136461503007651463, −9.972989736720247847015340861245, −9.05116620309363998969967452308, −8.02378468758446669433269821104, −7.0144279987015508599455437670, −5.58751895727689808514175587706, −3.47264189239668881615365296382, −2.29991144873989692608539961566, −1.267485586329476338372829650437, 1.7777362865297546605458641082, 3.23055681979799482798280129285, 5.14302894902692288342195006346, 5.94220050598389949232136018057, 7.48146478269382018501181940804, 8.60882607685397977040197243298, 9.52097091998449062798704970133, 10.303109879261373874111082739074, 11.27018843956938378657300415113, 13.54039545066582376179515608637, 14.10114321723724730163091335468, 15.19648632958608766672644963398, 16.28337532000538575661951157443, 16.82444864192200254111313123899, 18.36848078313903902678859123582, 18.77708513310971162282023091424, 20.38062383702351676401175209818, 20.932510430748718266958405225207, 22.07593407671576832325738423344, 23.29643083255276794570618028943, 24.56403064773589335377661257204, 25.433334239648763031884242671869, 26.10198295598842716282654103768, 26.80847986871314489335714210236, 27.83783973440049922683927748565

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