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Properties

Label 1-137-137.109-r0-0-0
Degree $1$
Conductor $137$
Sign $0.435 - 0.899i$
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  + (−0.932 − 0.361i)2-s + (0.961 − 0.273i)3-s + (0.739 + 0.673i)4-s + (−0.361 − 0.932i)5-s + (−0.995 − 0.0922i)6-s + (0.602 + 0.798i)7-s + (−0.445 − 0.895i)8-s + (0.850 − 0.526i)9-s + i·10-s + (−0.739 − 0.673i)11-s + (0.895 + 0.445i)12-s + (0.798 − 0.602i)13-s + (−0.273 − 0.961i)14-s + (−0.602 − 0.798i)15-s + (0.0922 + 0.995i)16-s + (−0.445 + 0.895i)17-s + ⋯
L(s)  = 1  + (−0.932 − 0.361i)2-s + (0.961 − 0.273i)3-s + (0.739 + 0.673i)4-s + (−0.361 − 0.932i)5-s + (−0.995 − 0.0922i)6-s + (0.602 + 0.798i)7-s + (−0.445 − 0.895i)8-s + (0.850 − 0.526i)9-s + i·10-s + (−0.739 − 0.673i)11-s + (0.895 + 0.445i)12-s + (0.798 − 0.602i)13-s + (−0.273 − 0.961i)14-s + (−0.602 − 0.798i)15-s + (0.0922 + 0.995i)16-s + (−0.445 + 0.895i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.435 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.435 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $0.435 - 0.899i$
Analytic conductor: \(0.636225\)
Root analytic conductor: \(0.636225\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (0:\ ),\ 0.435 - 0.899i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8310280565 - 0.5208731615i\)
\(L(\frac12)\) \(\approx\) \(0.8310280565 - 0.5208731615i\)
\(L(1)\) \(\approx\) \(0.8925614441 - 0.3344181564i\)
\(L(1)\) \(\approx\) \(0.8925614441 - 0.3344181564i\)

Euler product

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

−28.39908422388805550104243344994, −27.24074285423228084210846654083, −26.57096719911582374360490070609, −26.062564491309578405576860640936, −25.01890802720420182424770410536, −23.88670234141196250273397093517, −22.96160065183760676747081901285, −21.28116140515688333954474748121, −20.34382933607947897814763452600, −19.662261485455149978263112316051, −18.36787030379044536756591516769, −17.9504155401000060158244105932, −16.162486541120083374780745027234, −15.59123620496587754849306601721, −14.374209375278815024237782455257, −13.80377080756943371983284945678, −11.645576020346768303865441138874, −10.54591228685837407688947627990, −9.83249970126996496126320329003, −8.41597949172502668854043767074, −7.55812498045688165380411748488, −6.766532850036165312703543261794, −4.715494199765540803062453214891, −3.15257922116503038818387805184, −1.792300250791559839547152466064, 1.24393743423183929642827320492, 2.55512929702491107201040091547, 3.899329280566952423740698772035, 5.791332178632660562219711872008, 7.66506018553677582354754443316, 8.40455749608224679960094522846, 8.93088423584279533592637699701, 10.35466200670874038838182195770, 11.720949595581870110968456682, 12.652917451302060031437374293264, 13.70186848158309144320146773820, 15.56518215839784201694776422258, 15.77752228224191963182656705578, 17.46334030822018955915074595446, 18.390056579734682587437465997196, 19.23107103639680426893640566040, 20.29760094503812346143524199000, 20.85783519654356877277057612286, 21.803975771303862233493449230784, 23.97758366651499374447984446272, 24.46252145673512722710560850705, 25.46370371975417194078300268320, 26.36544119503619173417866935032, 27.38435595942657497679365949015, 28.204351168051800458932487590332

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