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Properties

Label 1-137-137.9-r0-0-0
Degree $1$
Conductor $137$
Sign $0.153 + 0.988i$
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 + (0.995 + 0.0922i)3-s + (−0.273 + 0.961i)4-s + (0.798 + 0.602i)5-s + (0.526 + 0.850i)6-s + (−0.739 − 0.673i)7-s + (−0.932 + 0.361i)8-s + (0.982 + 0.183i)9-s + i·10-s + (0.273 − 0.961i)11-s + (−0.361 + 0.932i)12-s + (−0.673 + 0.739i)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 + (0.995 + 0.0922i)3-s + (−0.273 + 0.961i)4-s + (0.798 + 0.602i)5-s + (0.526 + 0.850i)6-s + (−0.739 − 0.673i)7-s + (−0.932 + 0.361i)8-s + (0.982 + 0.183i)9-s + i·10-s + (0.273 − 0.961i)11-s + (−0.361 + 0.932i)12-s + (−0.673 + 0.739i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.153 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.153 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $0.153 + 0.988i$
Analytic conductor: \(0.636225\)
Root analytic conductor: \(0.636225\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (0:\ ),\ 0.153 + 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.465801613 + 1.255378724i\)
\(L(\frac12)\) \(\approx\) \(1.465801613 + 1.255378724i\)
\(L(1)\) \(\approx\) \(1.538357133 + 0.8791061977i\)
\(L(1)\) \(\approx\) \(1.538357133 + 0.8791061977i\)

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 \( 1 + (0.995 + 0.0922i)T \)
5 \( 1 + (0.798 + 0.602i)T \)
7 \( 1 + (-0.739 - 0.673i)T \)
11 \( 1 + (0.273 - 0.961i)T \)
13 \( 1 + (-0.673 + 0.739i)T \)
17 \( 1 + (-0.932 - 0.361i)T \)
19 \( 1 + (-0.445 - 0.895i)T \)
23 \( 1 + (0.526 - 0.850i)T \)
29 \( 1 + (-0.526 + 0.850i)T \)
31 \( 1 + (0.895 + 0.445i)T \)
37 \( 1 - T \)
41 \( 1 - iT \)
43 \( 1 + (-0.895 + 0.445i)T \)
47 \( 1 + (-0.183 + 0.982i)T \)
53 \( 1 + (0.895 - 0.445i)T \)
59 \( 1 + (-0.982 - 0.183i)T \)
61 \( 1 + (0.982 - 0.183i)T \)
67 \( 1 + (0.673 - 0.739i)T \)
71 \( 1 + (0.961 - 0.273i)T \)
73 \( 1 + (0.739 - 0.673i)T \)
79 \( 1 + (-0.995 + 0.0922i)T \)
83 \( 1 + (0.361 + 0.932i)T \)
89 \( 1 + (-0.798 - 0.602i)T \)
97 \( 1 + (-0.961 - 0.273i)T \)
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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

−28.44714193366659087899720756061, −27.555921569267594422243355880629, −26.14036006518835808288390136041, −24.93060601327478089869099282068, −24.69496443438747522945033863942, −23.07014994228999084707090816631, −22.01696906517554002199841560973, −21.19978020260621074345355202829, −20.24175159315210443155345785590, −19.56784252964647155630561921826, −18.506519125544364736274407160170, −17.30897255779015775757984174025, −15.46886488330795546056872440765, −14.87445550553067707164979677394, −13.519807429702993287006323026149, −12.89145802971748992247452269273, −12.09553269848704448903154210617, −10.07785918136235871333984219329, −9.61536394579658828324704471973, −8.52428397699866379266921962903, −6.6526849531327049432885084242, −5.34488213165289165212395572673, −4.044947446524116658774574020141, −2.62807317935978071314431395207, −1.76933754409114129232026029786, 2.499316980376972957810150423230, 3.48825818840050473740700163581, 4.79583577806173431483820718819, 6.61600207424105198674766000621, 6.9883878335832363865743429580, 8.66038706114789666992290790882, 9.482434193550329914783720908038, 10.89872067228035941900014979773, 12.7490007945815701845737080699, 13.702724476643109834150235823989, 14.137599112473460021278667742305, 15.250082607535853229270160717345, 16.35802253389298418244648135494, 17.31197838995929009024630646781, 18.66081629547079508071681664116, 19.67460620734007590780582261203, 21.00175618440738621172471213001, 21.851052995152477380551436310260, 22.576250824791448850552706814360, 24.07239303560711950522001850765, 24.73152617702937850634852405773, 25.862988064948869848314295460183, 26.39268394526706207670151552477, 27.022725220766877660532608626407, 29.16217815991903762935728238294

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