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Properties

Label 1-137-137.27-r1-0-0
Degree $1$
Conductor $137$
Sign $0.995 - 0.0946i$
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  + (0.183 + 0.982i)2-s + (0.990 + 0.138i)3-s + (−0.932 + 0.361i)4-s + (−0.565 − 0.824i)5-s + (0.0461 + 0.998i)6-s + (0.895 − 0.445i)7-s + (−0.526 − 0.850i)8-s + (0.961 + 0.273i)9-s + (0.707 − 0.707i)10-s + (−0.361 − 0.932i)11-s + (−0.973 + 0.228i)12-s + (−0.948 − 0.317i)13-s + (0.602 + 0.798i)14-s + (−0.445 − 0.895i)15-s + (0.739 − 0.673i)16-s + (0.526 − 0.850i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.982i)2-s + (0.990 + 0.138i)3-s + (−0.932 + 0.361i)4-s + (−0.565 − 0.824i)5-s + (0.0461 + 0.998i)6-s + (0.895 − 0.445i)7-s + (−0.526 − 0.850i)8-s + (0.961 + 0.273i)9-s + (0.707 − 0.707i)10-s + (−0.361 − 0.932i)11-s + (−0.973 + 0.228i)12-s + (−0.948 − 0.317i)13-s + (0.602 + 0.798i)14-s + (−0.445 − 0.895i)15-s + (0.739 − 0.673i)16-s + (0.526 − 0.850i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.995 - 0.0946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.995 - 0.0946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $0.995 - 0.0946i$
Analytic conductor: \(14.7226\)
Root analytic conductor: \(14.7226\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (1:\ ),\ 0.995 - 0.0946i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.255515177 - 0.1069750651i\)
\(L(\frac12)\) \(\approx\) \(2.255515177 - 0.1069750651i\)
\(L(1)\) \(\approx\) \(1.423695633 + 0.2674746263i\)
\(L(1)\) \(\approx\) \(1.423695633 + 0.2674746263i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad137 \( 1 \)
good2 \( 1 + (0.183 + 0.982i)T \)
3 \( 1 + (0.990 + 0.138i)T \)
5 \( 1 + (-0.565 - 0.824i)T \)
7 \( 1 + (0.895 - 0.445i)T \)
11 \( 1 + (-0.361 - 0.932i)T \)
13 \( 1 + (-0.948 - 0.317i)T \)
17 \( 1 + (0.526 - 0.850i)T \)
19 \( 1 + (0.995 + 0.0922i)T \)
23 \( 1 + (0.0461 - 0.998i)T \)
29 \( 1 + (0.998 + 0.0461i)T \)
31 \( 1 + (-0.769 - 0.638i)T \)
37 \( 1 + iT \)
41 \( 1 + (-0.707 - 0.707i)T \)
43 \( 1 + (0.638 + 0.769i)T \)
47 \( 1 + (-0.873 + 0.486i)T \)
53 \( 1 + (0.769 - 0.638i)T \)
59 \( 1 + (-0.273 + 0.961i)T \)
61 \( 1 + (0.961 - 0.273i)T \)
67 \( 1 + (-0.317 + 0.948i)T \)
71 \( 1 + (-0.914 + 0.403i)T \)
73 \( 1 + (0.445 - 0.895i)T \)
79 \( 1 + (-0.138 - 0.990i)T \)
83 \( 1 + (0.228 - 0.973i)T \)
89 \( 1 + (-0.824 + 0.565i)T \)
97 \( 1 + (0.403 - 0.914i)T \)
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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

−28.21016799559550325809240245689, −27.19705335731837261515199999013, −26.58810020642108886578649947399, −25.43293314081999413772106193382, −24.087153549197679673138004608330, −23.26520625573797733511850983866, −21.898763269763015983903338471312, −21.26412064104619453519879596126, −20.060653970212361305745769248383, −19.43636446913797046075981080357, −18.40219226784378330035557656084, −17.70220038526156883689739198732, −15.43362442808468399832236713713, −14.68221888082266263834308722446, −14.010462036342544658420164854787, −12.53956137337950786704744108681, −11.76520701837270970113259692574, −10.447711245514993210172618038751, −9.49938465036715914879401086566, −8.17424616547121112393897247353, −7.26721523664236787950369015728, −5.112039196588245316929417266714, −3.8757798255797715141270423687, −2.705313463063759891196736414079, −1.69212003489164449376568209492, 0.80126143295763003344151986829, 3.12799411488601692643792703872, 4.46474190312774140584453642235, 5.25250340530594647992031001639, 7.31201694165306180608610639587, 7.98928182527259666895531764225, 8.792516703255209254925773732998, 10.025178176082439828570699672057, 11.87175071049904444921729496234, 13.12319103733660980679536552077, 14.04967312547905335790508865497, 14.84605951499368267070429616118, 16.0001620720277396702522919292, 16.66331805274786421216007863328, 18.033608440491744799032126980604, 19.124880110529201295917192003992, 20.35746991772198163967708684556, 21.092015855071583699625761568657, 22.33259308272645848911912692542, 23.74004584855442444349154667504, 24.407495975093467015558179695274, 24.94353212331000751140686225147, 26.33113848351065715469893107544, 27.146661676284079602903642365871, 27.51555478631591880674061037276

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