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Properties

Label 1-671-671.120-r0-0-0
Degree $1$
Conductor $671$
Sign $0.458 + 0.888i$
Analytic cond. $3.11611$
Root an. cond. $3.11611$
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  + (0.994 + 0.104i)2-s + (0.809 + 0.587i)3-s + (0.978 + 0.207i)4-s + (−0.669 + 0.743i)5-s + (0.743 + 0.669i)6-s + (0.406 − 0.913i)7-s + (0.951 + 0.309i)8-s + (0.309 + 0.951i)9-s + (−0.743 + 0.669i)10-s + (0.669 + 0.743i)12-s + (0.5 + 0.866i)13-s + (0.5 − 0.866i)14-s + (−0.978 + 0.207i)15-s + (0.913 + 0.406i)16-s + (0.207 − 0.978i)17-s + (0.207 + 0.978i)18-s + ⋯
L(s)  = 1  + (0.994 + 0.104i)2-s + (0.809 + 0.587i)3-s + (0.978 + 0.207i)4-s + (−0.669 + 0.743i)5-s + (0.743 + 0.669i)6-s + (0.406 − 0.913i)7-s + (0.951 + 0.309i)8-s + (0.309 + 0.951i)9-s + (−0.743 + 0.669i)10-s + (0.669 + 0.743i)12-s + (0.5 + 0.866i)13-s + (0.5 − 0.866i)14-s + (−0.978 + 0.207i)15-s + (0.913 + 0.406i)16-s + (0.207 − 0.978i)17-s + (0.207 + 0.978i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(671\)    =    \(11 \cdot 61\)
Sign: $0.458 + 0.888i$
Analytic conductor: \(3.11611\)
Root analytic conductor: \(3.11611\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{671} (120, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 671,\ (0:\ ),\ 0.458 + 0.888i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.964713739 + 1.807504441i\)
\(L(\frac12)\) \(\approx\) \(2.964713739 + 1.807504441i\)
\(L(1)\) \(\approx\) \(2.235574280 + 0.8091820554i\)
\(L(1)\) \(\approx\) \(2.235574280 + 0.8091820554i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
61 \( 1 \)
good2 \( 1 + (0.994 + 0.104i)T \)
3 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (-0.669 + 0.743i)T \)
7 \( 1 + (0.406 - 0.913i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + (0.207 - 0.978i)T \)
19 \( 1 + (0.913 - 0.406i)T \)
23 \( 1 + (-0.951 + 0.309i)T \)
29 \( 1 + (-0.866 - 0.5i)T \)
31 \( 1 + (-0.994 + 0.104i)T \)
37 \( 1 + (0.587 + 0.809i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + (-0.207 - 0.978i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (0.951 + 0.309i)T \)
59 \( 1 + (0.994 + 0.104i)T \)
67 \( 1 + (-0.743 - 0.669i)T \)
71 \( 1 + (0.743 - 0.669i)T \)
73 \( 1 + (-0.669 - 0.743i)T \)
79 \( 1 + (0.207 + 0.978i)T \)
83 \( 1 + (0.104 - 0.994i)T \)
89 \( 1 + (-0.587 + 0.809i)T \)
97 \( 1 + (0.104 + 0.994i)T \)
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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

−22.699145928763051146209321738501, −21.73222324054188898757367879379, −20.878175677285571016323830171083, −20.24029209042743496851700501581, −19.69908221299321992572616220850, −18.70953670487175354247152052114, −17.9388154821302076034120313006, −16.51660609256451735199478547765, −15.74162733181772950422892062269, −14.95973501933031521278275978720, −14.45318447854482747758100593163, −13.23598740760361861186320876158, −12.72503858290614134663034169121, −12.04280746350130597632624112418, −11.27255317213408074690102124518, −9.95121146212104019714756076138, −8.69981426878370224331306490649, −8.05836520130478927014917492402, −7.29635971774182852729243944731, −5.93020995399833443583651931851, −5.33018589915125606381522262187, −3.964263183454985424444266866372, −3.38096875764993153908070879730, −2.142887558776612409579219202693, −1.282175931652958775735634345107, 1.71101839499106122045920359097, 2.88983618513241306076497501952, 3.71983200736271831791373246336, 4.277232902811529941664211509162, 5.27709396053969152913292920576, 6.7232128084382390823872074742, 7.454032276572425891158422267844, 8.05945425689214483301053103833, 9.48257719502255117603455718281, 10.46087989386470559165070315562, 11.29897173455655135635865321933, 11.82334996673811423301798483605, 13.44490693293279682866977543295, 13.8142323999279103099581207587, 14.55983103569232918086412882888, 15.26207401858425140284537679136, 16.15970655926997466793947077480, 16.60804321138751471224837854751, 18.1334802595434698823698183533, 19.09107218316437485215592900882, 20.11458449640201201854812833599, 20.357978525334535536316922693630, 21.33516419270847231888678923633, 22.15573891155108772833439647400, 22.77864799079307067927502177622

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