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Properties

Label 1-137-137.59-r0-0-0
Degree $1$
Conductor $137$
Sign $0.974 - 0.222i$
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  + (−0.602 − 0.798i)2-s + (0.0922 − 0.995i)3-s + (−0.273 + 0.961i)4-s + (−0.602 + 0.798i)5-s + (−0.850 + 0.526i)6-s + (0.739 + 0.673i)7-s + (0.932 − 0.361i)8-s + (−0.982 − 0.183i)9-s + 10-s + (−0.273 + 0.961i)11-s + (0.932 + 0.361i)12-s + (0.739 + 0.673i)13-s + (0.0922 − 0.995i)14-s + (0.739 + 0.673i)15-s + (−0.850 − 0.526i)16-s + (0.932 + 0.361i)17-s + ⋯
L(s)  = 1  + (−0.602 − 0.798i)2-s + (0.0922 − 0.995i)3-s + (−0.273 + 0.961i)4-s + (−0.602 + 0.798i)5-s + (−0.850 + 0.526i)6-s + (0.739 + 0.673i)7-s + (0.932 − 0.361i)8-s + (−0.982 − 0.183i)9-s + 10-s + (−0.273 + 0.961i)11-s + (0.932 + 0.361i)12-s + (0.739 + 0.673i)13-s + (0.0922 − 0.995i)14-s + (0.739 + 0.673i)15-s + (−0.850 − 0.526i)16-s + (0.932 + 0.361i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $0.974 - 0.222i$
Analytic conductor: \(0.636225\)
Root analytic conductor: \(0.636225\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (0:\ ),\ 0.974 - 0.222i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7333241888 - 0.08273824153i\)
\(L(\frac12)\) \(\approx\) \(0.7333241888 - 0.08273824153i\)
\(L(1)\) \(\approx\) \(0.7409004880 - 0.1919208631i\)
\(L(1)\) \(\approx\) \(0.7409004880 - 0.1919208631i\)

Euler product

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

−27.87115597096189078626277374942, −27.58447715416620440914662772859, −26.66055072393717939546481303763, −25.786744498445412127300449153703, −24.54550108632446769875130781096, −23.697114909737210296884205862560, −22.88161080187164723455481721377, −21.35197765237306883051096515745, −20.34917371720882031140860530219, −19.64933076917907355437262081456, −18.17021488968336880894691309187, −17.069418236132156514581960617427, −16.20859764777445720183202801265, −15.644335726884764319762822864717, −14.4130458740713951807333800595, −13.47749844288340787661432870069, −11.43450927831557834688255286935, −10.65408146142207974786954451704, −9.382855074851722542707770719534, −8.352435071818957241091475956779, −7.676437758727446966542425458934, −5.690134895399346897572717734189, −4.875239736736246614587256893226, −3.596002087014920233043330864341, −0.895637124416603890174012048431, 1.608967491449172491176178532041, 2.636204907662053969745357071466, 4.08918735446280854544297994273, 6.13119169389603920932926754549, 7.690042000873137346860686942086, 8.03208462520479591727900395406, 9.55489745666680296559388347917, 10.97983146873962021427343891099, 11.832619331021847813286915097325, 12.510757948683974949689918979240, 13.95842577688756234652600225278, 14.959754537513704089875352897315, 16.54365231765082735966776942110, 17.94008610254713634487207810245, 18.43478849499613909117922485049, 19.10895306292278583691883699454, 20.281258094564750570241451311557, 21.182518705717437506034825605146, 22.56645885043791212487327132355, 23.31010193571088085691646873800, 24.60840745303847554513018074521, 25.76469764478006445417086459111, 26.37092837581397519069905282305, 27.84275900689050844011207890230, 28.2835629544885680345678822340

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