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Properties

Label 1-137-137.41-r1-0-0
Degree $1$
Conductor $137$
Sign $-0.538 + 0.842i$
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  − i·2-s + (0.707 − 0.707i)3-s − 4-s + (−0.707 − 0.707i)5-s + (−0.707 − 0.707i)6-s − i·7-s + i·8-s − i·9-s + (−0.707 + 0.707i)10-s − i·11-s + (−0.707 + 0.707i)12-s + (0.707 − 0.707i)13-s − 14-s − 15-s + 16-s + i·17-s + ⋯
L(s)  = 1  − i·2-s + (0.707 − 0.707i)3-s − 4-s + (−0.707 − 0.707i)5-s + (−0.707 − 0.707i)6-s − i·7-s + i·8-s − i·9-s + (−0.707 + 0.707i)10-s − i·11-s + (−0.707 + 0.707i)12-s + (0.707 − 0.707i)13-s − 14-s − 15-s + 16-s + i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.538 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.538 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $-0.538 + 0.842i$
Analytic conductor: \(14.7226\)
Root analytic conductor: \(14.7226\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (1:\ ),\ -0.538 + 0.842i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.6528352117 - 1.191901703i\)
\(L(\frac12)\) \(\approx\) \(-0.6528352117 - 1.191901703i\)
\(L(1)\) \(\approx\) \(0.4596175059 - 0.9257457010i\)
\(L(1)\) \(\approx\) \(0.4596175059 - 0.9257457010i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad137 \( 1 \)
good2 \( 1 - iT \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
7 \( 1 - iT \)
11 \( 1 - iT \)
13 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 + iT \)
19 \( 1 + iT \)
23 \( 1 + (-0.707 + 0.707i)T \)
29 \( 1 + (-0.707 - 0.707i)T \)
31 \( 1 + (0.707 - 0.707i)T \)
37 \( 1 + iT \)
41 \( 1 + (0.707 + 0.707i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
47 \( 1 + (-0.707 - 0.707i)T \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + T \)
61 \( 1 + iT \)
67 \( 1 + (-0.707 - 0.707i)T \)
71 \( 1 + (0.707 - 0.707i)T \)
73 \( 1 + T \)
79 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 + (0.707 - 0.707i)T \)
89 \( 1 + (-0.707 + 0.707i)T \)
97 \( 1 + (-0.707 + 0.707i)T \)
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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

−28.22855873232970553906827698488, −27.784559317641591695041727691030, −26.59297131550630213646142405919, −25.982192138610638563143099675019, −25.181442868836086056683400503347, −24.08727094617571965574778417311, −22.79640738959689263530664415885, −22.20542287453858934437929460802, −21.071657640394974673193583099081, −19.680945269128518974158139351914, −18.68853087842710113316380182863, −17.85559345589494892436941456035, −16.05641599124967583346151956626, −15.74342220286092798882537365157, −14.7170058980518679122143535090, −14.04926636440692616577862151930, −12.52619328410422813997243621400, −11.06890991259057992407975801267, −9.60204369133559642535311378468, −8.82006987357495152004436795729, −7.68645959042750747217865247755, −6.614916410208976690131875401172, −5.01104582991356599015930272856, −4.00128805289255428184182677616, −2.59338583583351628727096739224, 0.5220665028234638885151800876, 1.52383783507553648312565026730, 3.42070389105211457374725789664, 4.00597928256367447111342744779, 5.913731593413781883936929604575, 7.98459518673400140806569395410, 8.24348075116783012172245023515, 9.71192015056197529120598172991, 11.00785698022076214524860014798, 12.074618739630014328213994265620, 13.178409479563002299438355489638, 13.66735412800156065267512925587, 15.00839043221587744641094643050, 16.5823886049198762768548883450, 17.69568949942480789301018982376, 18.9693351875251357647125089159, 19.547236260066104584867318887374, 20.4847442888601063061790351965, 21.04848062307625783641823379390, 22.731335679006155116711224367056, 23.651920744930012948844418449423, 24.290271789395318020319075126353, 25.82928785605061884489374542580, 26.77857976587623956406462266754, 27.60533961545026117346720859065

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