Properties

Label 1-139-139.20-r0-0-0
Degree $1$
Conductor $139$
Sign $0.817 - 0.576i$
Analytic cond. $0.645513$
Root an. cond. $0.645513$
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  + (−0.648 − 0.761i)2-s + (0.613 + 0.789i)3-s + (−0.158 + 0.987i)4-s + (0.538 − 0.842i)5-s + (0.203 − 0.979i)6-s + (0.746 − 0.665i)7-s + (0.854 − 0.519i)8-s + (−0.247 + 0.968i)9-s + (−0.990 + 0.136i)10-s + (−0.974 + 0.225i)11-s + (−0.877 + 0.480i)12-s + (0.377 − 0.926i)13-s + (−0.990 − 0.136i)14-s + (0.995 − 0.0909i)15-s + (−0.949 − 0.313i)16-s + (0.113 + 0.993i)17-s + ⋯
L(s)  = 1  + (−0.648 − 0.761i)2-s + (0.613 + 0.789i)3-s + (−0.158 + 0.987i)4-s + (0.538 − 0.842i)5-s + (0.203 − 0.979i)6-s + (0.746 − 0.665i)7-s + (0.854 − 0.519i)8-s + (−0.247 + 0.968i)9-s + (−0.990 + 0.136i)10-s + (−0.974 + 0.225i)11-s + (−0.877 + 0.480i)12-s + (0.377 − 0.926i)13-s + (−0.990 − 0.136i)14-s + (0.995 − 0.0909i)15-s + (−0.949 − 0.313i)16-s + (0.113 + 0.993i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.817 - 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.817 - 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(139\)
Sign: $0.817 - 0.576i$
Analytic conductor: \(0.645513\)
Root analytic conductor: \(0.645513\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{139} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 139,\ (0:\ ),\ 0.817 - 0.576i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.019869602 - 0.3236023975i\)
\(L(\frac12)\) \(\approx\) \(1.019869602 - 0.3236023975i\)
\(L(1)\) \(\approx\) \(1.001658906 - 0.2257161215i\)
\(L(1)\) \(\approx\) \(1.001658906 - 0.2257161215i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad139 \( 1 \)
good2 \( 1 + (-0.648 - 0.761i)T \)
3 \( 1 + (0.613 + 0.789i)T \)
5 \( 1 + (0.538 - 0.842i)T \)
7 \( 1 + (0.746 - 0.665i)T \)
11 \( 1 + (-0.974 + 0.225i)T \)
13 \( 1 + (0.377 - 0.926i)T \)
17 \( 1 + (0.113 + 0.993i)T \)
19 \( 1 + (0.803 - 0.595i)T \)
23 \( 1 + (0.203 + 0.979i)T \)
29 \( 1 + (0.934 - 0.356i)T \)
31 \( 1 + (-0.247 - 0.968i)T \)
37 \( 1 + (0.995 + 0.0909i)T \)
41 \( 1 + (-0.829 + 0.557i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (-0.998 + 0.0455i)T \)
53 \( 1 + (0.983 - 0.181i)T \)
59 \( 1 + (-0.775 + 0.631i)T \)
61 \( 1 + (-0.829 - 0.557i)T \)
67 \( 1 + (0.291 + 0.956i)T \)
71 \( 1 + (-0.998 - 0.0455i)T \)
73 \( 1 + (0.0227 + 0.999i)T \)
79 \( 1 + (-0.0682 + 0.997i)T \)
83 \( 1 + (0.291 - 0.956i)T \)
89 \( 1 + (-0.419 + 0.907i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−28.77120437754632197270261647683, −27.07785023493026008307503023959, −26.45673237875827083801670932237, −25.450604762059465885721955211584, −24.843440819042415919790073786786, −23.864667355984321926577157456, −22.95011723941905042457595672555, −21.42192499266609738058136162717, −20.36503717865789547860549822538, −18.8838154715277671913194587015, −18.36471628967476117715009863001, −17.902385268936554323716216461860, −16.33186083274742893223579758385, −15.125770350066310073947484629042, −14.21803990972927001837974613488, −13.64796599925520770862110042746, −11.877563788551183368680357086155, −10.609878444005561898777757265288, −9.314713714462719021560705670521, −8.35208701978195629393107797102, −7.32601216777571603082933605472, −6.365175922532507579939037802, −5.17955676299367176246874895155, −2.78970241800130730123324400297, −1.63578336842254283067396660257, 1.4022832704378011803198696479, 2.84826216081643360075193154551, 4.23646486539278225330008757690, 5.27473848789193854228361109492, 7.80951482286307131768031022974, 8.33474981087587966451204472361, 9.65122406290527003705880791504, 10.34078836642340838472321304667, 11.385467529705183591696320677271, 13.06676110450190398719369856800, 13.575058085151831504339540551219, 15.20365659951080279025787127061, 16.33609120667692777980108530789, 17.31673312189779598192642980112, 18.12865862493154097369854977115, 19.77692456514969001106529132795, 20.30564358831127812792005196915, 21.13599215748935617913205939567, 21.72453650045880590196860109644, 23.267386996102670497920917268652, 24.6930965149160201161462047753, 25.72536183837466846695910847927, 26.43765144494909353163859912290, 27.52204577149853516880320372564, 28.126150864524528786232960206311

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