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Properties

Label 1-2667-2667.1070-r1-0-0
Degree $1$
Conductor $2667$
Sign $-0.303 + 0.952i$
Analytic cond. $286.608$
Root an. cond. $286.608$
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  + (−0.623 + 0.781i)2-s + (−0.222 − 0.974i)4-s + (0.222 − 0.974i)5-s + (0.900 + 0.433i)8-s + (0.623 + 0.781i)10-s + (0.988 − 0.149i)11-s + (−0.955 + 0.294i)13-s + (−0.900 + 0.433i)16-s + (−0.733 − 0.680i)17-s − 19-s − 20-s + (−0.5 + 0.866i)22-s + (−0.988 − 0.149i)23-s + (−0.900 − 0.433i)25-s + (0.365 − 0.930i)26-s + ⋯
L(s)  = 1  + (−0.623 + 0.781i)2-s + (−0.222 − 0.974i)4-s + (0.222 − 0.974i)5-s + (0.900 + 0.433i)8-s + (0.623 + 0.781i)10-s + (0.988 − 0.149i)11-s + (−0.955 + 0.294i)13-s + (−0.900 + 0.433i)16-s + (−0.733 − 0.680i)17-s − 19-s − 20-s + (−0.5 + 0.866i)22-s + (−0.988 − 0.149i)23-s + (−0.900 − 0.433i)25-s + (0.365 − 0.930i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.303 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.303 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2667\)    =    \(3 \cdot 7 \cdot 127\)
Sign: $-0.303 + 0.952i$
Analytic conductor: \(286.608\)
Root analytic conductor: \(286.608\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2667} (1070, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2667,\ (1:\ ),\ -0.303 + 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1602431555 + 0.2193166563i\)
\(L(\frac12)\) \(\approx\) \(0.1602431555 + 0.2193166563i\)
\(L(1)\) \(\approx\) \(0.6548199193 + 0.01505056076i\)
\(L(1)\) \(\approx\) \(0.6548199193 + 0.01505056076i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
127 \( 1 \)
good2 \( 1 + (-0.623 + 0.781i)T \)
5 \( 1 + (0.222 - 0.974i)T \)
11 \( 1 + (0.988 - 0.149i)T \)
13 \( 1 + (-0.955 + 0.294i)T \)
17 \( 1 + (-0.733 - 0.680i)T \)
19 \( 1 - T \)
23 \( 1 + (-0.988 - 0.149i)T \)
29 \( 1 + (0.0747 - 0.997i)T \)
31 \( 1 + (0.733 - 0.680i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.733 - 0.680i)T \)
43 \( 1 + (-0.0747 + 0.997i)T \)
47 \( 1 + (-0.222 - 0.974i)T \)
53 \( 1 + (0.365 + 0.930i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.900 - 0.433i)T \)
67 \( 1 + (-0.826 - 0.563i)T \)
71 \( 1 + (0.988 + 0.149i)T \)
73 \( 1 + (-0.623 - 0.781i)T \)
79 \( 1 + (-0.988 - 0.149i)T \)
83 \( 1 + (-0.365 - 0.930i)T \)
89 \( 1 + (-0.623 + 0.781i)T \)
97 \( 1 + (0.0747 + 0.997i)T \)
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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

−19.123487906617076649622149367125, −18.17606792054842794743161973432, −17.56715666694771379397257849683, −17.218515629946895954687830441048, −16.287377789623163822490144219394, −15.33171894507497685502247790545, −14.57350458561293173193779963364, −13.99172743490683432870036880578, −13.02982871759325992764728293239, −12.33787937132006140459305599872, −11.66200542204296651601594850177, −10.91392478007183723309609321305, −10.234365983483909817430696984931, −9.78904570287078988704859743757, −8.78792127877241333366062080296, −8.237393066567361911634644681193, −7.04459865772115057671435753558, −6.83116011634876730780491012299, −5.714809551058203267895196806652, −4.44562759310982120975447013124, −3.831953669380568683964617987309, −2.92794860978091421707466393344, −2.14831853046459592861244837977, −1.490456838869121515742156492033, −0.07960458230903950792070227806, 0.60573996067904476490121549860, 1.69714940514603268488226896880, 2.36002996626445043200612338086, 4.089214690817800324728202725578, 4.57226144884432261500403820503, 5.378832417385976656895839263183, 6.33446990420857283739017785316, 6.75507009420496784074716194921, 7.86576018519620915776056056546, 8.43734476224523286863936288451, 9.18361480633863880130011609329, 9.70103432491104960723192647829, 10.39913903381445168880576163942, 11.623258113343699588754058269093, 12.006247641922301453991041697423, 13.181709501970294559952765128441, 13.743621645151572429922185195161, 14.48207475139187643473987721296, 15.24643018409389384290678116131, 15.93563485704153279306498256651, 16.69128892899142804512289118476, 17.221556605922473417392282380460, 17.5640720636584273911284951143, 18.64688954936550650799890737915, 19.29895277772150059153467488065

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