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Properties

Label 1-137-137.115-r0-0-0
Degree $1$
Conductor $137$
Sign $-0.428 - 0.903i$
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  + (−0.273 − 0.961i)2-s + (−0.982 + 0.183i)3-s + (−0.850 + 0.526i)4-s + (−0.273 + 0.961i)5-s + (0.445 + 0.895i)6-s + (0.0922 − 0.995i)7-s + (0.739 + 0.673i)8-s + (0.932 − 0.361i)9-s + 10-s + (−0.850 + 0.526i)11-s + (0.739 − 0.673i)12-s + (0.0922 − 0.995i)13-s + (−0.982 + 0.183i)14-s + (0.0922 − 0.995i)15-s + (0.445 − 0.895i)16-s + (0.739 − 0.673i)17-s + ⋯
L(s)  = 1  + (−0.273 − 0.961i)2-s + (−0.982 + 0.183i)3-s + (−0.850 + 0.526i)4-s + (−0.273 + 0.961i)5-s + (0.445 + 0.895i)6-s + (0.0922 − 0.995i)7-s + (0.739 + 0.673i)8-s + (0.932 − 0.361i)9-s + 10-s + (−0.850 + 0.526i)11-s + (0.739 − 0.673i)12-s + (0.0922 − 0.995i)13-s + (−0.982 + 0.183i)14-s + (0.0922 − 0.995i)15-s + (0.445 − 0.895i)16-s + (0.739 − 0.673i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.428 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.428 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $-0.428 - 0.903i$
Analytic conductor: \(0.636225\)
Root analytic conductor: \(0.636225\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (0:\ ),\ -0.428 - 0.903i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2593244661 - 0.4100335591i\)
\(L(\frac12)\) \(\approx\) \(0.2593244661 - 0.4100335591i\)
\(L(1)\) \(\approx\) \(0.5108337413 - 0.2713136662i\)
\(L(1)\) \(\approx\) \(0.5108337413 - 0.2713136662i\)

Euler product

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

−28.48012014658227747346976380333, −27.88128973425289440023092545320, −27.02183076639196702463068667738, −25.60693306632066584513356237078, −24.72483288306399243895216304003, −23.6507125579222700557279830109, −23.507623892086098885395401009455, −21.884446719225696541078830106706, −21.180665851431035004003859850821, −19.17411569354311911963026975748, −18.61783162910360350152046425807, −17.46017701871828036498126310250, −16.45812303205888931792460705290, −15.99471932035408126496450236195, −14.77424006239220418730840266773, −13.19677122290366186513156381961, −12.41444949199102458576753501717, −11.15900737232736388374227944830, −9.70062590530515077736290894469, −8.56794783431683964979967116153, −7.59135701576834999762674730846, −6.014617857424155473948604896846, −5.43881495439263808566574115925, −4.266142652697858175649093219777, −1.42654315151282058154797119957, 0.59687469371258502030319517006, 2.668557952303786788973275001627, 4.01612091942813903945465011386, 5.16352155691304245802441455227, 6.93447459094581102821852578155, 7.89421234975718424936219037321, 9.88486517931984077449185766841, 10.55690308329024987803727015950, 11.17340050710632559927336807318, 12.42762847482136964589976310333, 13.39982725928593587518019566288, 14.81907448169211429796324529937, 16.153101009442304811969289475618, 17.417236312516814499192757838920, 18.027884234330011692385328509742, 18.97887589310241722510877356945, 20.23696830848186130572308722517, 21.13664366972514951193645419868, 22.26470620444065223051182951616, 23.09068579549867686860517492060, 23.49857073293397143852633216904, 25.56258591052032951912424081817, 26.72816549163071891802749572049, 27.17916442522405559341072389958, 28.250710280451354430713062196186

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