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Properties

Label 1-137-137.46-r1-0-0
Degree $1$
Conductor $137$
Sign $0.00369 - 0.999i$
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  + (0.895 − 0.445i)2-s + (0.998 − 0.0461i)3-s + (0.602 − 0.798i)4-s + (−0.948 + 0.317i)5-s + (0.873 − 0.486i)6-s + (−0.361 − 0.932i)7-s + (0.183 − 0.982i)8-s + (0.995 − 0.0922i)9-s + (−0.707 + 0.707i)10-s + (0.798 + 0.602i)11-s + (0.565 − 0.824i)12-s + (0.403 − 0.914i)13-s + (−0.739 − 0.673i)14-s + (−0.932 + 0.361i)15-s + (−0.273 − 0.961i)16-s + (−0.183 − 0.982i)17-s + ⋯
L(s)  = 1  + (0.895 − 0.445i)2-s + (0.998 − 0.0461i)3-s + (0.602 − 0.798i)4-s + (−0.948 + 0.317i)5-s + (0.873 − 0.486i)6-s + (−0.361 − 0.932i)7-s + (0.183 − 0.982i)8-s + (0.995 − 0.0922i)9-s + (−0.707 + 0.707i)10-s + (0.798 + 0.602i)11-s + (0.565 − 0.824i)12-s + (0.403 − 0.914i)13-s + (−0.739 − 0.673i)14-s + (−0.932 + 0.361i)15-s + (−0.273 − 0.961i)16-s + (−0.183 − 0.982i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00369 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00369 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $0.00369 - 0.999i$
Analytic conductor: \(14.7226\)
Root analytic conductor: \(14.7226\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (1:\ ),\ 0.00369 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.604341465 - 2.594737439i\)
\(L(\frac12)\) \(\approx\) \(2.604341465 - 2.594737439i\)
\(L(1)\) \(\approx\) \(1.949299675 - 0.9857432431i\)
\(L(1)\) \(\approx\) \(1.949299675 - 0.9857432431i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad137 \( 1 \)
good2 \( 1 + (0.895 - 0.445i)T \)
3 \( 1 + (0.998 - 0.0461i)T \)
5 \( 1 + (-0.948 + 0.317i)T \)
7 \( 1 + (-0.361 - 0.932i)T \)
11 \( 1 + (0.798 + 0.602i)T \)
13 \( 1 + (0.403 - 0.914i)T \)
17 \( 1 + (-0.183 - 0.982i)T \)
19 \( 1 + (-0.526 - 0.850i)T \)
23 \( 1 + (0.873 + 0.486i)T \)
29 \( 1 + (-0.486 + 0.873i)T \)
31 \( 1 + (-0.973 + 0.228i)T \)
37 \( 1 + iT \)
41 \( 1 + (0.707 + 0.707i)T \)
43 \( 1 + (-0.228 + 0.973i)T \)
47 \( 1 + (0.638 - 0.769i)T \)
53 \( 1 + (0.973 + 0.228i)T \)
59 \( 1 + (0.0922 + 0.995i)T \)
61 \( 1 + (0.995 + 0.0922i)T \)
67 \( 1 + (-0.914 - 0.403i)T \)
71 \( 1 + (-0.990 - 0.138i)T \)
73 \( 1 + (0.932 + 0.361i)T \)
79 \( 1 + (0.0461 - 0.998i)T \)
83 \( 1 + (-0.824 + 0.565i)T \)
89 \( 1 + (0.317 + 0.948i)T \)
97 \( 1 + (-0.138 - 0.990i)T \)
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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

−28.51381209711417965203716207434, −27.23258935641526201913506469405, −26.29729040186691710713239473021, −25.30786877783040232998657734474, −24.511032193420589008561233882913, −23.743745025371725216449442002187, −22.49139483079412949791446737435, −21.487094686579716257621253564172, −20.69112419614985069003448434269, −19.41709707275983761518883039545, −18.87480181597615973575607229736, −16.80349573268672545085136870180, −15.97111003179058390750327833903, −15.04524370929130512324966900, −14.35407511300056753955245141874, −13.01850315236145953258077879133, −12.26176825435056252167085531105, −11.10230420134058171584195761507, −8.97914249061376425859205542982, −8.43719288856660663057047999845, −7.12916902866693168707603784379, −5.90948267265877867550284149756, −4.17121701781978632038038426474, −3.5796331861339633514961655603, −2.05346256050119165854307918493, 1.04255442205342936969170009240, 2.87380615724621922796163502442, 3.70710340385170222830730458727, 4.68328005815237674936050953567, 6.8283645014992417790215710076, 7.412499380293021362464563448697, 9.11527065779550538134217649086, 10.36923498595067358703576452916, 11.386046883656960131751202279651, 12.737395518504781692336334899507, 13.486805187669155256208049953045, 14.66636538769308744807230843743, 15.28189805338540371105695676185, 16.3683926390068185389589868442, 18.258718451694069394261658178471, 19.5635381344211665221165664288, 19.92659512830021399340884987528, 20.6970235629408678258106899302, 22.1276098311351894804860694236, 23.01299852423963695188317405295, 23.798102775941963222227476760091, 24.97383722590346335151077894145, 25.85251227613654500786115957669, 27.148428059984378081481556172687, 27.82261832011955718528762467959

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