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Properties

Label 1-127-127.18-r0-0-0
Degree $1$
Conductor $127$
Sign $-0.105 + 0.994i$
Analytic cond. $0.589785$
Root an. cond. $0.589785$
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  + (−0.222 + 0.974i)2-s + (−0.853 − 0.521i)3-s + (−0.900 − 0.433i)4-s + (0.826 + 0.563i)5-s + (0.698 − 0.715i)6-s + (−0.969 − 0.246i)7-s + (0.623 − 0.781i)8-s + (0.456 + 0.889i)9-s + (−0.733 + 0.680i)10-s + (0.921 − 0.388i)11-s + (0.542 + 0.840i)12-s + (0.270 + 0.962i)13-s + (0.456 − 0.889i)14-s + (−0.411 − 0.911i)15-s + (0.623 + 0.781i)16-s + (−0.411 + 0.911i)17-s + ⋯
L(s)  = 1  + (−0.222 + 0.974i)2-s + (−0.853 − 0.521i)3-s + (−0.900 − 0.433i)4-s + (0.826 + 0.563i)5-s + (0.698 − 0.715i)6-s + (−0.969 − 0.246i)7-s + (0.623 − 0.781i)8-s + (0.456 + 0.889i)9-s + (−0.733 + 0.680i)10-s + (0.921 − 0.388i)11-s + (0.542 + 0.840i)12-s + (0.270 + 0.962i)13-s + (0.456 − 0.889i)14-s + (−0.411 − 0.911i)15-s + (0.623 + 0.781i)16-s + (−0.411 + 0.911i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.105 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.105 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(127\)
Sign: $-0.105 + 0.994i$
Analytic conductor: \(0.589785\)
Root analytic conductor: \(0.589785\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{127} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 127,\ (0:\ ),\ -0.105 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4564098632 + 0.5072399689i\)
\(L(\frac12)\) \(\approx\) \(0.4564098632 + 0.5072399689i\)
\(L(1)\) \(\approx\) \(0.6441638180 + 0.3446261025i\)
\(L(1)\) \(\approx\) \(0.6441638180 + 0.3446261025i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad127 \( 1 \)
good2 \( 1 + (-0.222 + 0.974i)T \)
3 \( 1 + (-0.853 - 0.521i)T \)
5 \( 1 + (0.826 + 0.563i)T \)
7 \( 1 + (-0.969 - 0.246i)T \)
11 \( 1 + (0.921 - 0.388i)T \)
13 \( 1 + (0.270 + 0.962i)T \)
17 \( 1 + (-0.411 + 0.911i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.921 + 0.388i)T \)
29 \( 1 + (-0.661 + 0.749i)T \)
31 \( 1 + (0.995 + 0.0995i)T \)
37 \( 1 + (-0.939 - 0.342i)T \)
41 \( 1 + (0.995 - 0.0995i)T \)
43 \( 1 + (-0.318 - 0.947i)T \)
47 \( 1 + (0.0747 + 0.997i)T \)
53 \( 1 + (0.542 - 0.840i)T \)
59 \( 1 + (0.766 - 0.642i)T \)
61 \( 1 + (0.365 - 0.930i)T \)
67 \( 1 + (-0.0249 + 0.999i)T \)
71 \( 1 + (-0.797 + 0.603i)T \)
73 \( 1 + (0.955 + 0.294i)T \)
79 \( 1 + (-0.124 - 0.992i)T \)
83 \( 1 + (-0.998 - 0.0498i)T \)
89 \( 1 + (-0.733 - 0.680i)T \)
97 \( 1 + (0.980 - 0.198i)T \)
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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

−28.3516313662066060983962147310, −28.08646988473237903442354689973, −26.868782802574581421778167793109, −25.75320236316947725457614893869, −24.63978065211722265710325480894, −22.808361138581031568346280428632, −22.54835182505510861744453420277, −21.44872263577414138504840902870, −20.59241738350901689513499185650, −19.5669642617707879304824741250, −18.19384390802910791659489084564, −17.35910242623683328372237167451, −16.60136874062534046254011406617, −15.24786711443555678637884117697, −13.481169189850363880286784961, −12.70175833603056076421863781636, −11.715400161093852392400142226554, −10.48691440268662551859890003088, −9.56895684824748793780175846674, −8.911678776246614626364937751814, −6.65749910282714924506348659784, −5.37900140830261835203832202789, −4.26241501162332590444780563529, −2.72979040046181108355533335897, −0.85036128541527488107753619910, 1.529231383490776667039284305644, 3.92353271128780571232374679368, 5.64918021107229725547876418251, 6.485082613451362504390859776735, 7.00347117823264388387514361368, 8.798369082604567523252509132014, 9.94590485061156290449372628999, 10.991088803784847404897878091233, 12.67417212392833218209133089249, 13.58897143769786416610782312179, 14.52257326998381261227185297532, 16.030538329475347222688895191432, 16.961857055719773557055526767690, 17.48063421947315203090073679193, 18.9276939769508402025135075894, 19.16142162934392821958299780091, 21.52804032830560501185983567722, 22.3782562362351790989388652421, 23.08719156802363262693560596905, 24.14709909927143770258205555295, 25.120471033015401781067249477548, 25.9235013139046856770004932401, 26.9131486660960295588110279119, 28.140015041711082542976099321836, 29.055737327733615937852702712147

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