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Properties

Label 1-671-671.175-r0-0-0
Degree $1$
Conductor $671$
Sign $-0.991 + 0.128i$
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 + (0.809 + 0.587i)5-s + (−0.587 + 0.809i)6-s + (−0.951 + 0.309i)7-s + (0.587 + 0.809i)8-s + (−0.809 − 0.587i)9-s + (0.587 + 0.809i)10-s + (−0.809 + 0.587i)12-s − 13-s − 14-s + (−0.809 + 0.587i)15-s + (0.309 + 0.951i)16-s + (−0.587 + 0.809i)17-s + (−0.587 − 0.809i)18-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)2-s + (−0.309 + 0.951i)3-s + (0.809 + 0.587i)4-s + (0.809 + 0.587i)5-s + (−0.587 + 0.809i)6-s + (−0.951 + 0.309i)7-s + (0.587 + 0.809i)8-s + (−0.809 − 0.587i)9-s + (0.587 + 0.809i)10-s + (−0.809 + 0.587i)12-s − 13-s − 14-s + (−0.809 + 0.587i)15-s + (0.309 + 0.951i)16-s + (−0.587 + 0.809i)17-s + (−0.587 − 0.809i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(671\)    =    \(11 \cdot 61\)
Sign: $-0.991 + 0.128i$
Analytic conductor: \(3.11611\)
Root analytic conductor: \(3.11611\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{671} (175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 671,\ (0:\ ),\ -0.991 + 0.128i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1218918579 + 1.884051216i\)
\(L(\frac12)\) \(\approx\) \(0.1218918579 + 1.884051216i\)
\(L(1)\) \(\approx\) \(1.079050885 + 1.083548670i\)
\(L(1)\) \(\approx\) \(1.079050885 + 1.083548670i\)

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 \( 1 + (0.809 + 0.587i)T \)
7 \( 1 + (-0.951 + 0.309i)T \)
13 \( 1 - T \)
17 \( 1 + (-0.587 + 0.809i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (-0.587 + 0.809i)T \)
29 \( 1 - iT \)
31 \( 1 + (-0.951 + 0.309i)T \)
37 \( 1 + (-0.951 + 0.309i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + (0.587 + 0.809i)T \)
47 \( 1 + T \)
53 \( 1 + (0.587 + 0.809i)T \)
59 \( 1 + (0.951 + 0.309i)T \)
67 \( 1 + (0.587 - 0.809i)T \)
71 \( 1 + (-0.587 - 0.809i)T \)
73 \( 1 + (0.809 - 0.587i)T \)
79 \( 1 + (-0.587 - 0.809i)T \)
83 \( 1 + (-0.309 + 0.951i)T \)
89 \( 1 + (0.951 + 0.309i)T \)
97 \( 1 + (-0.309 - 0.951i)T \)
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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

−22.33959877038451249545660152987, −21.94334352581187008490006519775, −20.50421425861098677718667386087, −20.19325218675844118482268316810, −19.22579789868888414107777285630, −18.44000306388542046999900136114, −17.36702198535625843418084625839, −16.52945169088328525632558373414, −15.94628100208293606621400432426, −14.35589415009397476004950292406, −13.98879269441102206502262692632, −13.00873694197083970095789478619, −12.54414336050354160421212184894, −11.90891552893480390718012159990, −10.64382564939539798305919930521, −9.91050285054938452017786195766, −8.84159869430785834479088003368, −7.308429244216908290554814177436, −6.79177241550868468206435595668, −5.737995561713224677901092840101, −5.22202108879931912209385316497, −3.9380311546893096778589889448, −2.61357596506216149560466479663, −1.95387511854434387623969114673, −0.62674335548312468679221592476, 2.22609174460354575404972992203, 3.006234951859316098275536015932, 3.92451293340711315711534620706, 5.00574136700398319866336019, 5.83822864438615714391769946631, 6.44767314557227041350204030550, 7.41957805589592470933767046509, 8.95577989621758952574106633667, 9.759936549066407657045452544814, 10.5711583781825039494578087115, 11.45095782252943069595831422191, 12.38581063110917593198989730955, 13.308401393442193534575760650874, 14.0946260560732881843766290051, 15.07899774774229891378863441866, 15.46319459479023996922695776413, 16.43515236669891093413302606785, 17.23007176963516800023503279642, 17.850303222674033760367561561279, 19.40624403906244493766228871948, 20.021368073869651646673596025911, 21.19202354314762864451935966876, 21.82465154341218423345945138909, 22.18241726154214455024026626434, 22.82505424318925066275604902286

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