Properties

Label 1-137-137.13-r1-0-0
Degree $1$
Conductor $137$
Sign $-0.992 - 0.122i$
Analytic cond. $14.7226$
Root an. cond. $14.7226$
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  + (0.526 − 0.850i)2-s + (0.403 + 0.914i)3-s + (−0.445 − 0.895i)4-s + (0.228 − 0.973i)5-s + (0.990 + 0.138i)6-s + (−0.183 − 0.982i)7-s + (−0.995 − 0.0922i)8-s + (−0.673 + 0.739i)9-s + (−0.707 − 0.707i)10-s + (−0.895 + 0.445i)11-s + (0.638 − 0.769i)12-s + (−0.824 − 0.565i)13-s + (−0.932 − 0.361i)14-s + (0.982 − 0.183i)15-s + (−0.602 + 0.798i)16-s + (0.995 − 0.0922i)17-s + ⋯
L(s)  = 1  + (0.526 − 0.850i)2-s + (0.403 + 0.914i)3-s + (−0.445 − 0.895i)4-s + (0.228 − 0.973i)5-s + (0.990 + 0.138i)6-s + (−0.183 − 0.982i)7-s + (−0.995 − 0.0922i)8-s + (−0.673 + 0.739i)9-s + (−0.707 − 0.707i)10-s + (−0.895 + 0.445i)11-s + (0.638 − 0.769i)12-s + (−0.824 − 0.565i)13-s + (−0.932 − 0.361i)14-s + (0.982 − 0.183i)15-s + (−0.602 + 0.798i)16-s + (0.995 − 0.0922i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.992 - 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.992 - 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $-0.992 - 0.122i$
Analytic conductor: \(14.7226\)
Root analytic conductor: \(14.7226\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (1:\ ),\ -0.992 - 0.122i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08736914270 - 1.418651269i\)
\(L(\frac12)\) \(\approx\) \(0.08736914270 - 1.418651269i\)
\(L(1)\) \(\approx\) \(0.9692357912 - 0.6984880333i\)
\(L(1)\) \(\approx\) \(0.9692357912 - 0.6984880333i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad137 \( 1 \)
good2 \( 1 + (0.526 - 0.850i)T \)
3 \( 1 + (0.403 + 0.914i)T \)
5 \( 1 + (0.228 - 0.973i)T \)
7 \( 1 + (-0.183 - 0.982i)T \)
11 \( 1 + (-0.895 + 0.445i)T \)
13 \( 1 + (-0.824 - 0.565i)T \)
17 \( 1 + (0.995 - 0.0922i)T \)
19 \( 1 + (-0.961 + 0.273i)T \)
23 \( 1 + (0.990 - 0.138i)T \)
29 \( 1 + (-0.138 - 0.990i)T \)
31 \( 1 + (-0.873 + 0.486i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.707 - 0.707i)T \)
43 \( 1 + (0.486 - 0.873i)T \)
47 \( 1 + (-0.998 + 0.0461i)T \)
53 \( 1 + (0.873 + 0.486i)T \)
59 \( 1 + (0.739 + 0.673i)T \)
61 \( 1 + (-0.673 - 0.739i)T \)
67 \( 1 + (0.565 - 0.824i)T \)
71 \( 1 + (0.948 - 0.317i)T \)
73 \( 1 + (-0.982 - 0.183i)T \)
79 \( 1 + (0.914 + 0.403i)T \)
83 \( 1 + (0.769 - 0.638i)T \)
89 \( 1 + (0.973 + 0.228i)T \)
97 \( 1 + (0.317 - 0.948i)T \)
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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

−29.17251411262847640786598174058, −27.44562671495873873263365848934, −26.12702562697226387012370507426, −25.74888602532052715917885428350, −24.77876049614216864522090798484, −23.823883171658978053527069400352, −23.00072485800075661989470874681, −21.83260536028855419100794158532, −21.12499589333011549952171256556, −19.19505812422156871512458787169, −18.59696449925716366343752992577, −17.70416904095447210487624032613, −16.44539681706944609944184539366, −14.90357211482693160671772192197, −14.66477785009891207653973215416, −13.3306868197994596633592374470, −12.54973820473562165423925848424, −11.36112284548726609149193829145, −9.472584818697743441568586506134, −8.28671917157970244342865347843, −7.25767050946062155903836736665, −6.329844422468070275816392876547, −5.315825266568793740447058577024, −3.23111168118396461603912824119, −2.41261463893637737258098290093, 0.414491061792006381042500875492, 2.26560879794932028201577873404, 3.67557796547231394388454024762, 4.7173275270398080191212039541, 5.52348799932273566451852901469, 7.75599761028762290494807115710, 9.16242632165054690247616979317, 10.09584133739540751597746873852, 10.7492610819190163797218685590, 12.41242014770011684676815945536, 13.189679182879150680365037626259, 14.278868187870561206493625587926, 15.275761713946710097575367916686, 16.51826350575890252738657371604, 17.48073530577482687914233553533, 19.226595984602729063921537682823, 20.05104233163984721092020123372, 20.875498993343863251626403629967, 21.32005245397974884515919295836, 22.76051107179576768236560132024, 23.42503339734611405592633214915, 24.72932795536113592644331490729, 25.89589963102189737450659273824, 27.142887637867535083034581353232, 27.76476815194577434139937914178

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