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Properties

Label 1-133-133.107-r1-0-0
Degree $1$
Conductor $133$
Sign $-0.469 + 0.882i$
Analytic cond. $14.2928$
Root an. cond. $14.2928$
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  − 2-s + (0.5 + 0.866i)3-s + 4-s + 5-s + (−0.5 − 0.866i)6-s − 8-s + (−0.5 + 0.866i)9-s − 10-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)12-s + (0.5 + 0.866i)13-s + (0.5 + 0.866i)15-s + 16-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)18-s + ⋯
L(s)  = 1  − 2-s + (0.5 + 0.866i)3-s + 4-s + 5-s + (−0.5 − 0.866i)6-s − 8-s + (−0.5 + 0.866i)9-s − 10-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)12-s + (0.5 + 0.866i)13-s + (0.5 + 0.866i)15-s + 16-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.469 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.469 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $-0.469 + 0.882i$
Analytic conductor: \(14.2928\)
Root analytic conductor: \(14.2928\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 133,\ (1:\ ),\ -0.469 + 0.882i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7058855531 + 1.175309219i\)
\(L(\frac12)\) \(\approx\) \(0.7058855531 + 1.175309219i\)
\(L(1)\) \(\approx\) \(0.8330280454 + 0.4433451230i\)
\(L(1)\) \(\approx\) \(0.8330280454 + 0.4433451230i\)

Euler product

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

−28.354069784808133843898490827226, −26.69607031233818128803126790792, −26.11735343958197831906028600024, −25.0523539765298843846434195719, −24.58084738433232237720209335173, −23.406798033167735076643186432414, −21.67313170485889273520050901893, −20.73444903089732916833687003524, −19.796963060801563260906123904689, −18.734028461599712767989923942088, −17.97876446699219570182621531358, −17.235566720417169226062668711831, −15.907111144865688729961020461, −14.62553258434143938933260906960, −13.448316524987463398497652338799, −12.52603313391776594428958656652, −11.01574505534965831667382879497, −10.01453594138933398649180024501, −8.67479067325327974637980474297, −8.082831180513851830069757146461, −6.55625555663482546324909189344, −5.83865949900599474804340592399, −3.08901546627084844258241391393, −2.01510492683497910985360365948, −0.68406036945237382488549564323, 1.773145505350909677619408070090, 2.87596615845942148419023058637, 4.731693384817567148884829712600, 6.18382883791823198260760570643, 7.565639325238854061155383297706, 8.937752003491870251745872737124, 9.60726716075645193750745581975, 10.451946095877102338375734165223, 11.60791264485177659065069348596, 13.33908901361964791429527384929, 14.47637994310207430419456721982, 15.6220345028947088121380457098, 16.45286783778003514969053888971, 17.554384912425515424715368190109, 18.41927441584943990742016655038, 19.713788033704352192193747909623, 20.701563281298558701243589812248, 21.241674149841273061500957148517, 22.39816967505542648196411159654, 24.037114052841645371779704446286, 25.32553777056727669034928963409, 25.77664699803654671776175965216, 26.58837012222332937545028215986, 27.68906749228959845430226431221, 28.53091195552334117677438486001

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