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Properties

Label 1-137-137.99-r0-0-0
Degree $1$
Conductor $137$
Sign $0.203 + 0.979i$
Analytic cond. $0.636225$
Root an. cond. $0.636225$
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  + (0.739 + 0.673i)2-s + (0.850 − 0.526i)3-s + (0.0922 + 0.995i)4-s + (−0.739 + 0.673i)5-s + (0.982 + 0.183i)6-s + (−0.273 + 0.961i)7-s + (−0.602 + 0.798i)8-s + (0.445 − 0.895i)9-s − 10-s + (0.0922 + 0.995i)11-s + (0.602 + 0.798i)12-s + (0.273 − 0.961i)13-s + (−0.850 + 0.526i)14-s + (−0.273 + 0.961i)15-s + (−0.982 + 0.183i)16-s + (−0.602 − 0.798i)17-s + ⋯
L(s)  = 1  + (0.739 + 0.673i)2-s + (0.850 − 0.526i)3-s + (0.0922 + 0.995i)4-s + (−0.739 + 0.673i)5-s + (0.982 + 0.183i)6-s + (−0.273 + 0.961i)7-s + (−0.602 + 0.798i)8-s + (0.445 − 0.895i)9-s − 10-s + (0.0922 + 0.995i)11-s + (0.602 + 0.798i)12-s + (0.273 − 0.961i)13-s + (−0.850 + 0.526i)14-s + (−0.273 + 0.961i)15-s + (−0.982 + 0.183i)16-s + (−0.602 − 0.798i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.203 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.203 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $0.203 + 0.979i$
Analytic conductor: \(0.636225\)
Root analytic conductor: \(0.636225\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (0:\ ),\ 0.203 + 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.368121834 + 1.113442728i\)
\(L(\frac12)\) \(\approx\) \(1.368121834 + 1.113442728i\)
\(L(1)\) \(\approx\) \(1.477126276 + 0.7279037932i\)
\(L(1)\) \(\approx\) \(1.477126276 + 0.7279037932i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad137 \( 1 \)
good2 \( 1 + (0.739 + 0.673i)T \)
3 \( 1 + (0.850 - 0.526i)T \)
5 \( 1 + (-0.739 + 0.673i)T \)
7 \( 1 + (-0.273 + 0.961i)T \)
11 \( 1 + (0.0922 + 0.995i)T \)
13 \( 1 + (0.273 - 0.961i)T \)
17 \( 1 + (-0.602 - 0.798i)T \)
19 \( 1 + (0.932 - 0.361i)T \)
23 \( 1 + (0.982 - 0.183i)T \)
29 \( 1 + (0.982 - 0.183i)T \)
31 \( 1 + (-0.932 - 0.361i)T \)
37 \( 1 + T \)
41 \( 1 - T \)
43 \( 1 + (-0.932 + 0.361i)T \)
47 \( 1 + (-0.445 + 0.895i)T \)
53 \( 1 + (-0.932 + 0.361i)T \)
59 \( 1 + (0.445 - 0.895i)T \)
61 \( 1 + (0.445 + 0.895i)T \)
67 \( 1 + (0.273 - 0.961i)T \)
71 \( 1 + (-0.0922 + 0.995i)T \)
73 \( 1 + (-0.273 - 0.961i)T \)
79 \( 1 + (0.850 + 0.526i)T \)
83 \( 1 + (0.602 - 0.798i)T \)
89 \( 1 + (-0.739 + 0.673i)T \)
97 \( 1 + (-0.0922 - 0.995i)T \)
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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

−28.44838681573870454688230949503, −27.07660208427564147621451208124, −26.74183795701047136933911527825, −25.1195661662220968100317120637, −24.02816420169026658745804857588, −23.40600562455230709032108728485, −22.0270769360525609105045314901, −21.1816326670610016165440368198, −20.21629394806103984694453425870, −19.63166536875689599950966868979, −18.806766356219583196785803961869, −16.61526739626171636719602606330, −15.94265653913724596857982443739, −14.73922559306500541954669594082, −13.71426001495765836360852672129, −13.05165129366934867479417633024, −11.567466661758900401142594077758, −10.662409754684088156942786405806, −9.410881143223618520748550339783, −8.3878477669749397913147282400, −6.84632521162926804867330191533, −5.03537528924165262531700860745, −3.95903301764857811444524324648, −3.31721240870681302892047884371, −1.388640227001250436350047404553, 2.576513372533156683338791787306, 3.30302352577390161175958591978, 4.84164846015503208993934464319, 6.45993573271264865981062830872, 7.31952891570948131018441854885, 8.259719245292012606557672741109, 9.45638098626274644838785221632, 11.46707401998238756873332010878, 12.433320281663910751711409004932, 13.30135710691816020088863447439, 14.5938137455883129575189961301, 15.256054819282198175848363338966, 15.89314790688386004872501776122, 17.83506116856774966467258344192, 18.42652244433133143933545341580, 19.82112599945751392415952471035, 20.61542979980854633503528248345, 22.063445083851570724194000723456, 22.8027615233077307589859638329, 23.7529692174825331774432433052, 25.04320218562446049519202836009, 25.30400888262680711063277490382, 26.46844338622490220546831715721, 27.32398694208079480319600680704, 28.943201306494967645088704024000

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