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Properties

Label 1-137-137.57-r1-0-0
Degree $1$
Conductor $137$
Sign $-0.441 + 0.897i$
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  + (−0.961 + 0.273i)2-s + (−0.565 + 0.824i)3-s + (0.850 − 0.526i)4-s + (0.873 − 0.486i)5-s + (0.317 − 0.948i)6-s + (−0.995 − 0.0922i)7-s + (−0.673 + 0.739i)8-s + (−0.361 − 0.932i)9-s + (−0.707 + 0.707i)10-s + (0.526 + 0.850i)11-s + (−0.0461 + 0.998i)12-s + (−0.638 − 0.769i)13-s + (0.982 − 0.183i)14-s + (−0.0922 + 0.995i)15-s + (0.445 − 0.895i)16-s + (0.673 + 0.739i)17-s + ⋯
L(s)  = 1  + (−0.961 + 0.273i)2-s + (−0.565 + 0.824i)3-s + (0.850 − 0.526i)4-s + (0.873 − 0.486i)5-s + (0.317 − 0.948i)6-s + (−0.995 − 0.0922i)7-s + (−0.673 + 0.739i)8-s + (−0.361 − 0.932i)9-s + (−0.707 + 0.707i)10-s + (0.526 + 0.850i)11-s + (−0.0461 + 0.998i)12-s + (−0.638 − 0.769i)13-s + (0.982 − 0.183i)14-s + (−0.0922 + 0.995i)15-s + (0.445 − 0.895i)16-s + (0.673 + 0.739i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.441 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.441 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $-0.441 + 0.897i$
Analytic conductor: \(14.7226\)
Root analytic conductor: \(14.7226\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (1:\ ),\ -0.441 + 0.897i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3925109342 + 0.6303381623i\)
\(L(\frac12)\) \(\approx\) \(0.3925109342 + 0.6303381623i\)
\(L(1)\) \(\approx\) \(0.5712209706 + 0.2383309058i\)
\(L(1)\) \(\approx\) \(0.5712209706 + 0.2383309058i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad137 \( 1 \)
good2 \( 1 + (-0.961 + 0.273i)T \)
3 \( 1 + (-0.565 + 0.824i)T \)
5 \( 1 + (0.873 - 0.486i)T \)
7 \( 1 + (-0.995 - 0.0922i)T \)
11 \( 1 + (0.526 + 0.850i)T \)
13 \( 1 + (-0.638 - 0.769i)T \)
17 \( 1 + (0.673 + 0.739i)T \)
19 \( 1 + (0.798 - 0.602i)T \)
23 \( 1 + (0.317 + 0.948i)T \)
29 \( 1 + (-0.948 + 0.317i)T \)
31 \( 1 + (0.138 + 0.990i)T \)
37 \( 1 + iT \)
41 \( 1 + (0.707 + 0.707i)T \)
43 \( 1 + (-0.990 - 0.138i)T \)
47 \( 1 + (-0.914 - 0.403i)T \)
53 \( 1 + (-0.138 + 0.990i)T \)
59 \( 1 + (0.932 - 0.361i)T \)
61 \( 1 + (-0.361 + 0.932i)T \)
67 \( 1 + (-0.769 + 0.638i)T \)
71 \( 1 + (-0.973 + 0.228i)T \)
73 \( 1 + (0.0922 + 0.995i)T \)
79 \( 1 + (-0.824 + 0.565i)T \)
83 \( 1 + (0.998 - 0.0461i)T \)
89 \( 1 + (-0.486 - 0.873i)T \)
97 \( 1 + (0.228 - 0.973i)T \)
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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

−28.19860111129817696985071964750, −26.848647979886547099875605962811, −26.01166450598993783340926908435, −24.94666547423707144404829463561, −24.43327869518845364177550184609, −22.680633167136622616881665128591, −22.04181887859920575539345618211, −20.80737898179947520506994585308, −19.31273313306338798946684231526, −18.84676872530553673738388243595, −17.989272080796935868399420724052, −16.71472069531593744974232424540, −16.44133065864997914183167123178, −14.418176815440859374634482916957, −13.26167137659020055211268356328, −12.13758016714228930198354241326, −11.21378326625773750566872225239, −9.980694422908187471255959241233, −9.14434553323916529255834161395, −7.523780808547993583888668990347, −6.5944159755689755625088902252, −5.77849307638064635879715154482, −3.15404030452941081960559091084, −1.97444890002640360058260993389, −0.47528204122264926626813262219, 1.20071081764592211762227643607, 3.11470966567418544576629903381, 5.08023422028951250302090385575, 6.01324864012842657548167991086, 7.1608749329665514871589671334, 8.92592261531343800851610142816, 9.82374524835182627303860134928, 10.188940365386289571685471180191, 11.75479251904966394056718742160, 12.8780741347588128541398857999, 14.63459784352497728088739216369, 15.58034067146422692506779646642, 16.641136330615218315005450418872, 17.24892062184574338766053062543, 18.05789197482372451343648911368, 19.66885408088241277744174229514, 20.34028763647084960365507285586, 21.533514647029553191361930594047, 22.502409050090764875674938724362, 23.65353332272079443135623283296, 25.0064318050882373530078424295, 25.690685327703599759033648033510, 26.571203449017108652951327839666, 27.76137880756020831720003843577, 28.335375067684440411547923346609

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