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Properties

Label 1-137-137.31-r1-0-0
Degree $1$
Conductor $137$
Sign $0.966 - 0.255i$
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  + (−0.673 + 0.739i)2-s + (−0.973 − 0.228i)3-s + (−0.0922 − 0.995i)4-s + (−0.0461 + 0.998i)5-s + (0.824 − 0.565i)6-s + (−0.961 − 0.273i)7-s + (0.798 + 0.602i)8-s + (0.895 + 0.445i)9-s + (−0.707 − 0.707i)10-s + (−0.995 + 0.0922i)11-s + (−0.138 + 0.990i)12-s + (−0.873 + 0.486i)13-s + (0.850 − 0.526i)14-s + (0.273 − 0.961i)15-s + (−0.982 + 0.183i)16-s + (−0.798 + 0.602i)17-s + ⋯
L(s)  = 1  + (−0.673 + 0.739i)2-s + (−0.973 − 0.228i)3-s + (−0.0922 − 0.995i)4-s + (−0.0461 + 0.998i)5-s + (0.824 − 0.565i)6-s + (−0.961 − 0.273i)7-s + (0.798 + 0.602i)8-s + (0.895 + 0.445i)9-s + (−0.707 − 0.707i)10-s + (−0.995 + 0.0922i)11-s + (−0.138 + 0.990i)12-s + (−0.873 + 0.486i)13-s + (0.850 − 0.526i)14-s + (0.273 − 0.961i)15-s + (−0.982 + 0.183i)16-s + (−0.798 + 0.602i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.966 - 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.966 - 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $0.966 - 0.255i$
Analytic conductor: \(14.7226\)
Root analytic conductor: \(14.7226\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (1:\ ),\ 0.966 - 0.255i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3209367161 - 0.04174794246i\)
\(L(\frac12)\) \(\approx\) \(0.3209367161 - 0.04174794246i\)
\(L(1)\) \(\approx\) \(0.3942816564 + 0.1394397502i\)
\(L(1)\) \(\approx\) \(0.3942816564 + 0.1394397502i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad137 \( 1 \)
good2 \( 1 + (-0.673 + 0.739i)T \)
3 \( 1 + (-0.973 - 0.228i)T \)
5 \( 1 + (-0.0461 + 0.998i)T \)
7 \( 1 + (-0.961 - 0.273i)T \)
11 \( 1 + (-0.995 + 0.0922i)T \)
13 \( 1 + (-0.873 + 0.486i)T \)
17 \( 1 + (-0.798 + 0.602i)T \)
19 \( 1 + (-0.361 - 0.932i)T \)
23 \( 1 + (0.824 + 0.565i)T \)
29 \( 1 + (0.565 - 0.824i)T \)
31 \( 1 + (0.403 + 0.914i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.707 - 0.707i)T \)
43 \( 1 + (0.914 + 0.403i)T \)
47 \( 1 + (0.317 + 0.948i)T \)
53 \( 1 + (-0.403 + 0.914i)T \)
59 \( 1 + (0.445 - 0.895i)T \)
61 \( 1 + (0.895 - 0.445i)T \)
67 \( 1 + (-0.486 - 0.873i)T \)
71 \( 1 + (0.769 - 0.638i)T \)
73 \( 1 + (-0.273 - 0.961i)T \)
79 \( 1 + (-0.228 - 0.973i)T \)
83 \( 1 + (-0.990 + 0.138i)T \)
89 \( 1 + (-0.998 - 0.0461i)T \)
97 \( 1 + (0.638 - 0.769i)T \)
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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

−28.505173624733456622504648932981, −27.46057396312186603594722164050, −26.73303151784776059860174243512, −25.412918616910877267184036991377, −24.3726010427780670318179149024, −23.017570035617972185559150958081, −22.20704574778057489789219164759, −21.15779657916942004560984899055, −20.32893589314700426612151521908, −19.1523406223144855404588016628, −18.17305467149148881076670413832, −17.104029881131801034631185541926, −16.36182907840155325989037936686, −15.556852940698863485784655651582, −13.07534346450346981371216628978, −12.66018988353487877219072510301, −11.65731595341307952811683941268, −10.35751404134948954304367876113, −9.63847987287709248017104147457, −8.42719248660602609671296168159, −7.00922096921252192923732367912, −5.43676623579361895055958266539, −4.29992819313819099129149692785, −2.62725457962710251226209619480, −0.72954303134322704753784433929, 0.28328795942060035420064037697, 2.39796967233019308380287725631, 4.590892437246149784849882628385, 5.99068981441010877242818641799, 6.8511560651438564461853543806, 7.54303576867472122299629888857, 9.39720199287389750475540648071, 10.45213669280841333166562278569, 11.08255188703177952493933422352, 12.74191829759022401342418945986, 13.87890300092572008239605039095, 15.37366415931449062999527206152, 15.93329611659272887115542050665, 17.28459933061292056760720201604, 17.78677517094035361865253594001, 19.11645035903423986969859574062, 19.41060633959183495533987442793, 21.54121711501841617072315096589, 22.57932891597700171316829957571, 23.31067301354318752147316308340, 24.10140978613476060632223386736, 25.384331141453391098208459484129, 26.42643523427663799909562478398, 26.881197314801160536416267758170, 28.32511801769917812227870140770

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