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Properties

Label 1-1339-1339.1128-r0-0-0
Degree $1$
Conductor $1339$
Sign $-0.988 - 0.151i$
Analytic cond. $6.21828$
Root an. cond. $6.21828$
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  + (−0.0922 + 0.995i)2-s + (0.213 + 0.976i)3-s + (−0.982 − 0.183i)4-s + (0.998 + 0.0615i)5-s + (−0.992 + 0.122i)6-s + (0.273 + 0.961i)7-s + (0.273 − 0.961i)8-s + (−0.908 + 0.417i)9-s + (−0.153 + 0.988i)10-s + (0.908 − 0.417i)11-s + (−0.0307 − 0.999i)12-s + (−0.982 + 0.183i)14-s + (0.153 + 0.988i)15-s + (0.932 + 0.361i)16-s + (−0.602 + 0.798i)17-s + (−0.332 − 0.943i)18-s + ⋯
L(s)  = 1  + (−0.0922 + 0.995i)2-s + (0.213 + 0.976i)3-s + (−0.982 − 0.183i)4-s + (0.998 + 0.0615i)5-s + (−0.992 + 0.122i)6-s + (0.273 + 0.961i)7-s + (0.273 − 0.961i)8-s + (−0.908 + 0.417i)9-s + (−0.153 + 0.988i)10-s + (0.908 − 0.417i)11-s + (−0.0307 − 0.999i)12-s + (−0.982 + 0.183i)14-s + (0.153 + 0.988i)15-s + (0.932 + 0.361i)16-s + (−0.602 + 0.798i)17-s + (−0.332 − 0.943i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1339\)    =    \(13 \cdot 103\)
Sign: $-0.988 - 0.151i$
Analytic conductor: \(6.21828\)
Root analytic conductor: \(6.21828\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1339} (1128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1339,\ (0:\ ),\ -0.988 - 0.151i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1284413216 + 1.689394723i\)
\(L(\frac12)\) \(\approx\) \(-0.1284413216 + 1.689394723i\)
\(L(1)\) \(\approx\) \(0.6782663088 + 0.9943157936i\)
\(L(1)\) \(\approx\) \(0.6782663088 + 0.9943157936i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
103 \( 1 \)
good2 \( 1 + (-0.0922 + 0.995i)T \)
3 \( 1 + (0.213 + 0.976i)T \)
5 \( 1 + (0.998 + 0.0615i)T \)
7 \( 1 + (0.273 + 0.961i)T \)
11 \( 1 + (0.908 - 0.417i)T \)
17 \( 1 + (-0.602 + 0.798i)T \)
19 \( 1 + (0.952 - 0.303i)T \)
23 \( 1 + (-0.908 - 0.417i)T \)
29 \( 1 + (0.445 + 0.895i)T \)
31 \( 1 + (-0.932 - 0.361i)T \)
37 \( 1 + (0.0307 + 0.999i)T \)
41 \( 1 + (-0.445 + 0.895i)T \)
43 \( 1 + (0.881 - 0.473i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.952 + 0.303i)T \)
59 \( 1 + (0.696 + 0.717i)T \)
61 \( 1 + (0.992 + 0.122i)T \)
67 \( 1 + (-0.969 + 0.243i)T \)
71 \( 1 + (-0.445 + 0.895i)T \)
73 \( 1 + (-0.445 + 0.895i)T \)
79 \( 1 + (0.445 + 0.895i)T \)
83 \( 1 + (0.696 - 0.717i)T \)
89 \( 1 + (-0.332 + 0.943i)T \)
97 \( 1 + (-0.992 + 0.122i)T \)
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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

−20.540680684574160529739340798563, −19.78813441562882679841597512508, −19.17644651672400799750081235078, −18.03236126083402421235412297068, −17.77221844650170781041282941900, −17.221535831949259495370654476209, −16.23075008398801059658687857096, −14.51048309482906819914383231952, −14.06881316686152016880236808658, −13.61845409180526385526297201073, −12.82165044584698010124917369516, −12.026176115433466501178954976070, −11.33143898311704627580874027649, −10.436594396368765157877819081925, −9.480452263853110347835160940467, −9.091317609619111495024188055705, −7.88959222224189641636935598830, −7.20703869285325022463068208093, −6.20764383081146043219864332502, −5.2453377153252988601412443096, −4.19747297927060522220487899728, −3.25911872931790458985427518037, −2.15851556706462827032948857191, −1.5902431131768816750945975771, −0.68938719093566959043326806654, 1.44452352762505813301247951422, 2.66270182352408121287318481504, 3.74745353874685927127751701570, 4.68460160974874686069253195889, 5.53594560950795586495990847796, 6.001927230100645332652603750780, 6.90202277452252527958180030361, 8.30684410727983272558337839698, 8.77628165617905421076568810229, 9.42699035541049558846907060988, 10.09522441798566606093465220915, 11.044426823516017179530183353964, 12.08380603838946565759156742490, 13.19332259179198393719703001297, 14.00331601766356365818037193952, 14.565395159598581742922375739051, 15.14195425015421410859420445869, 16.029840096199567495709792034053, 16.61393444854110005340490695158, 17.42594152063928945316690821872, 18.00452836472790487156619877860, 18.82397318660055840788313811543, 19.81212015990913423985026822565, 20.65944802248089136427988124748, 21.84258702164652273186078400990

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