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Properties

Label 1-127-127.44-r0-0-0
Degree $1$
Conductor $127$
Sign $-0.906 + 0.421i$
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  + (0.623 + 0.781i)2-s + (−0.411 − 0.911i)3-s + (−0.222 + 0.974i)4-s + (−0.733 + 0.680i)5-s + (0.456 − 0.889i)6-s + (−0.998 + 0.0498i)7-s + (−0.900 + 0.433i)8-s + (−0.661 + 0.749i)9-s + (−0.988 − 0.149i)10-s + (−0.853 + 0.521i)11-s + (0.980 − 0.198i)12-s + (0.542 + 0.840i)13-s + (−0.661 − 0.749i)14-s + (0.921 + 0.388i)15-s + (−0.900 − 0.433i)16-s + (0.921 − 0.388i)17-s + ⋯
L(s)  = 1  + (0.623 + 0.781i)2-s + (−0.411 − 0.911i)3-s + (−0.222 + 0.974i)4-s + (−0.733 + 0.680i)5-s + (0.456 − 0.889i)6-s + (−0.998 + 0.0498i)7-s + (−0.900 + 0.433i)8-s + (−0.661 + 0.749i)9-s + (−0.988 − 0.149i)10-s + (−0.853 + 0.521i)11-s + (0.980 − 0.198i)12-s + (0.542 + 0.840i)13-s + (−0.661 − 0.749i)14-s + (0.921 + 0.388i)15-s + (−0.900 − 0.433i)16-s + (0.921 − 0.388i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.906 + 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.906 + 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(127\)
Sign: $-0.906 + 0.421i$
Analytic conductor: \(0.589785\)
Root analytic conductor: \(0.589785\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{127} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 127,\ (0:\ ),\ -0.906 + 0.421i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1322597498 + 0.5985871460i\)
\(L(\frac12)\) \(\approx\) \(0.1322597498 + 0.5985871460i\)
\(L(1)\) \(\approx\) \(0.6730287144 + 0.4306037719i\)
\(L(1)\) \(\approx\) \(0.6730287144 + 0.4306037719i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad127 \( 1 \)
good2 \( 1 + (0.623 + 0.781i)T \)
3 \( 1 + (-0.411 - 0.911i)T \)
5 \( 1 + (-0.733 + 0.680i)T \)
7 \( 1 + (-0.998 + 0.0498i)T \)
11 \( 1 + (-0.853 + 0.521i)T \)
13 \( 1 + (0.542 + 0.840i)T \)
17 \( 1 + (0.921 - 0.388i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.853 - 0.521i)T \)
29 \( 1 + (0.698 + 0.715i)T \)
31 \( 1 + (-0.797 + 0.603i)T \)
37 \( 1 + (0.766 - 0.642i)T \)
41 \( 1 + (-0.797 - 0.603i)T \)
43 \( 1 + (-0.969 + 0.246i)T \)
47 \( 1 + (0.955 - 0.294i)T \)
53 \( 1 + (0.980 + 0.198i)T \)
59 \( 1 + (0.173 + 0.984i)T \)
61 \( 1 + (0.0747 + 0.997i)T \)
67 \( 1 + (-0.583 + 0.811i)T \)
71 \( 1 + (0.878 - 0.478i)T \)
73 \( 1 + (0.365 + 0.930i)T \)
79 \( 1 + (-0.0249 + 0.999i)T \)
83 \( 1 + (-0.318 - 0.947i)T \)
89 \( 1 + (-0.988 + 0.149i)T \)
97 \( 1 + (0.270 + 0.962i)T \)
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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

−28.34732397708682278408717657968, −27.79519328379788612671771370368, −26.71931853985333539738558878792, −25.46923487273336566973165861018, −23.66870796321682894958725938143, −23.39007527446373688613497269909, −22.23979816144052742045089298839, −21.35785644682056427955261547071, −20.37604468434131285213113306572, −19.6368449810131403580171471011, −18.43603413456149613566445607120, −16.80996618323151904049476130719, −15.74122013802836579322375525297, −15.231337115865399712065748613236, −13.492356964557793059627017071672, −12.61255576600091393843061392979, −11.55054231563126717991287265405, −10.51532451513323953512370660983, −9.61392255651534710206857287508, −8.29967567561864986041476845606, −6.10699809254191074480168982532, −5.18178888770408229414756118389, −3.90716452907647423003791039463, −3.08206571092706087202051006166, −0.46913928780731564855660926186, 2.612723481332616149228842657718, 3.91539464958742841859276461404, 5.59021831526479791294122875193, 6.65794108470266097030104864293, 7.37522926211476049778338368246, 8.47899136711087537307283954388, 10.41781982468145510413940958923, 11.93314884408502669826033081048, 12.532490625671695925676839745875, 13.697308273393688893021700219715, 14.67791707272619565581596679571, 16.0433659083247574144749917503, 16.5767086511181224256161831292, 18.235152688503830406513939169232, 18.665808531686637642639461389682, 20.02269466808633412514452361053, 21.58369079424306887815758508111, 22.74408930937052374543849029516, 23.274923045600964072849655315552, 23.87778816986313191992281786733, 25.4239575271126036052898493376, 25.78975145702811035694759336054, 27.010235114279519387803182366427, 28.41461393099508960725807346740, 29.50116236353955813123552044825

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