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Properties

Label 1-137-137.98-r0-0-0
Degree $1$
Conductor $137$
Sign $-0.369 - 0.929i$
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  + (0.850 − 0.526i)2-s + (−0.361 − 0.932i)3-s + (0.445 − 0.895i)4-s + (0.526 − 0.850i)5-s + (−0.798 − 0.602i)6-s + (0.982 + 0.183i)7-s + (−0.0922 − 0.995i)8-s + (−0.739 + 0.673i)9-s − i·10-s + (−0.445 + 0.895i)11-s + (−0.995 − 0.0922i)12-s + (−0.183 + 0.982i)13-s + (0.932 − 0.361i)14-s + (−0.982 − 0.183i)15-s + (−0.602 − 0.798i)16-s + (−0.0922 + 0.995i)17-s + ⋯
L(s)  = 1  + (0.850 − 0.526i)2-s + (−0.361 − 0.932i)3-s + (0.445 − 0.895i)4-s + (0.526 − 0.850i)5-s + (−0.798 − 0.602i)6-s + (0.982 + 0.183i)7-s + (−0.0922 − 0.995i)8-s + (−0.739 + 0.673i)9-s − i·10-s + (−0.445 + 0.895i)11-s + (−0.995 − 0.0922i)12-s + (−0.183 + 0.982i)13-s + (0.932 − 0.361i)14-s + (−0.982 − 0.183i)15-s + (−0.602 − 0.798i)16-s + (−0.0922 + 0.995i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.369 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.369 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $-0.369 - 0.929i$
Analytic conductor: \(0.636225\)
Root analytic conductor: \(0.636225\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (98, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (0:\ ),\ -0.369 - 0.929i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9521801630 - 1.403251049i\)
\(L(\frac12)\) \(\approx\) \(0.9521801630 - 1.403251049i\)
\(L(1)\) \(\approx\) \(1.229636932 - 0.9930239716i\)
\(L(1)\) \(\approx\) \(1.229636932 - 0.9930239716i\)

Euler product

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

−29.159411425849633874050223375201, −27.37072766011926037552674046917, −26.84095324108613319554730877413, −25.82581271826247622203254921839, −24.77985983302902043906743742472, −23.67530756585596811125782520712, −22.60326714742755543933682359593, −22.03405561803539736877328472761, −21.026350260392019885873398278607, −20.40665308301178096966941785790, −18.271036696212974001617325912573, −17.49732309961956495360508652549, −16.36936501000410765722305575468, −15.4411120318088265534714682596, −14.38408598661748142853166683493, −13.88079340236350843396163366084, −12.16877828722364141237800424198, −11.03764112163629837347120029753, −10.336744631069305753924854027299, −8.57182481715359151323586062904, −7.33637273374736435522590455758, −5.8283565982829672949262753987, −5.23273368855771772135354029560, −3.78205935404568490393340074591, −2.652136213025971187811837318745, 1.52379829179776721008515893705, 2.20290425511755926539316669681, 4.49349097728435088359663540741, 5.29631171984679554711861749994, 6.46236930427074898485285901349, 7.83370020451553174944554503356, 9.32001163447182937950411811648, 10.79655349979045015926415671503, 11.889121711773314227838693947829, 12.58815992082886195154141933748, 13.585359592721407915068990240718, 14.4216186522774822842730760216, 15.81277863931557009162468839238, 17.28650719143751450979521800915, 18.008944049158013630455258446732, 19.33485127014140252896511788838, 20.2465921923360333156483521826, 21.27231945639354400022623286046, 22.05992319726049213345841118897, 23.59227335528128956473610527505, 23.92951773200729867831848294490, 24.77421295589153702485408730520, 25.827699056534861714487966312, 27.87011119680052609985306596265, 28.43589590684128869098372452164

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