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Properties

Label 1-671-671.150-r0-0-0
Degree $1$
Conductor $671$
Sign $0.738 - 0.673i$
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  + (−0.951 − 0.309i)2-s + (−0.309 − 0.951i)3-s + (0.809 + 0.587i)4-s − 5-s + i·6-s + (−0.951 − 0.309i)7-s + (−0.587 − 0.809i)8-s + (−0.809 + 0.587i)9-s + (0.951 + 0.309i)10-s + (0.309 − 0.951i)12-s + (−0.309 − 0.951i)13-s + (0.809 + 0.587i)14-s + (0.309 + 0.951i)15-s + (0.309 + 0.951i)16-s + i·17-s + (0.951 − 0.309i)18-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)2-s + (−0.309 − 0.951i)3-s + (0.809 + 0.587i)4-s − 5-s + i·6-s + (−0.951 − 0.309i)7-s + (−0.587 − 0.809i)8-s + (−0.809 + 0.587i)9-s + (0.951 + 0.309i)10-s + (0.309 − 0.951i)12-s + (−0.309 − 0.951i)13-s + (0.809 + 0.587i)14-s + (0.309 + 0.951i)15-s + (0.309 + 0.951i)16-s + i·17-s + (0.951 − 0.309i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.738 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.738 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(671\)    =    \(11 \cdot 61\)
Sign: $0.738 - 0.673i$
Analytic conductor: \(3.11611\)
Root analytic conductor: \(3.11611\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{671} (150, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 671,\ (0:\ ),\ 0.738 - 0.673i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3321103141 - 0.1287165542i\)
\(L(\frac12)\) \(\approx\) \(0.3321103141 - 0.1287165542i\)
\(L(1)\) \(\approx\) \(0.3952428363 - 0.1585861413i\)
\(L(1)\) \(\approx\) \(0.3952428363 - 0.1585861413i\)

Euler product

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

−22.71747975626347068542782506926, −22.256059721786038503557965072083, −21.06333515898127376982452545633, −20.17937722706257685238272002572, −19.609160995392238401985955565900, −18.79271123865163784440249952401, −17.94861195806243594571538550090, −16.82976094635752670507500299344, −16.12765011857662616399812930474, −15.87415966608576104197279141532, −14.96722145176238583714312700438, −14.1151119282644723875377850700, −12.46839137970858871276739612158, −11.6056298854555519435420861466, −11.10112724187855484692617072956, −9.97603044851077562744654021971, −9.28234042406680458332473149366, −8.70448059613897331218396891559, −7.40238261355711978404069798603, −6.70468396416317830550937971252, −5.62944456531290216137757767462, −4.58454261030200275714948035669, −3.458691061447591860069909522397, −2.46479458971710179610588300918, −0.46115298852108920681300110821, 0.59762957372788545425079739705, 1.85038663834958844221159254895, 3.10922766538141385864598431854, 3.84917407190597820967607264315, 5.71417502762859696368931224895, 6.5366529443623576678115183189, 7.629140399491610932598012890539, 7.84733785585809533562159861013, 8.95152792069297784733457113734, 10.17889210100465349455850753687, 10.81156724257401494340513888656, 11.853277627212959957529747322384, 12.50398560401739105085420085708, 13.006005234690943397842264643578, 14.41521552226829423450000825034, 15.59403094482742322803415138119, 16.25630851542330978817737776836, 17.07199829498295316687416147051, 17.839415185888131912998723052128, 18.740936881426311929981857699604, 19.39318937063649349260779814036, 19.82631348408469537967073340778, 20.59265752283572312767472322137, 22.01050786405914368186736392072, 22.76412267534321633095053751186

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