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Properties

Label 1-1169-1169.333-r1-0-0
Degree $1$
Conductor $1169$
Sign $0.895 + 0.444i$
Analytic cond. $125.626$
Root an. cond. $125.626$
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  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)5-s + 6-s + 8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s − 13-s − 15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.5 − 0.866i)18-s + (−0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)5-s + 6-s + 8-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s − 13-s − 15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.5 − 0.866i)18-s + (−0.5 + 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1169 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1169 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1169\)    =    \(7 \cdot 167\)
Sign: $0.895 + 0.444i$
Analytic conductor: \(125.626\)
Root analytic conductor: \(125.626\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1169} (333, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1169,\ (1:\ ),\ 0.895 + 0.444i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7525569845 + 0.1762949633i\)
\(L(\frac12)\) \(\approx\) \(0.7525569845 + 0.1762949633i\)
\(L(1)\) \(\approx\) \(0.6353170310 - 0.04031733613i\)
\(L(1)\) \(\approx\) \(0.6353170310 - 0.04031733613i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
167 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 - T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 - T \)
43 \( 1 - T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 - T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + T \)
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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

−21.25359816003052937360882327961, −20.212383861684808191520837229021, −19.632110108794448629753705895216, −18.53948562711846671139560422572, −17.91764846753657115265661830125, −17.33563305907067500830202081560, −16.67013187451852973856399565719, −15.55767867712869177342781809722, −14.88770811037707305380534778946, −13.96940357517773301253772846260, −13.03374575562795405877819402808, −12.05453446418173927248973777016, −11.43415464604829629645128529183, −10.62453147298262774304279111815, −9.81380145208510466650081886251, −9.708154218741172964909211932021, −8.51974383746314033759291905636, −7.28786891201200649636947299481, −6.69732890227240935187516512369, −5.084692467830468831392455189725, −4.84621364645989945999755021822, −3.41325514883346243626261207240, −2.83341351621336777931202739711, −1.8331206662793982464701083121, −0.31987755096635941552143140526, 0.618654355908734090544029348259, 1.47447707606016446843359042096, 2.49716198626592566371149323734, 4.31776602724321882658356404287, 5.26086796265586070900396873857, 5.836151628299670730442377689529, 6.51416496910188078736784137887, 7.60248914858776361207149567158, 8.24782167401486953809816891477, 8.85707038527822360027911061181, 10.073298954039653993296192612067, 10.59255476015966047971502459094, 11.829274641156256908638063774152, 12.75017251756102786123037603842, 13.23702267615201062098485928978, 14.17818776505995314338216744067, 14.83118045861352297120888022356, 16.09372595431462333389972257873, 16.81216710281008047443829949520, 16.98975706509279423933835091763, 17.950834124901479558404489235998, 18.66707592621895082375215709800, 19.31147289052059806703840099175, 20.04154556844217069473170702402, 21.21027325478898817585071613366

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