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Properties

Label 1-927-927.22-r1-0-0
Degree $1$
Conductor $927$
Sign $-0.943 + 0.331i$
Analytic cond. $99.6199$
Root an. cond. $99.6199$
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  + (−0.696 − 0.717i)2-s + (−0.0307 + 0.999i)4-s + (−0.332 − 0.943i)5-s + (−0.952 + 0.303i)7-s + (0.739 − 0.673i)8-s + (−0.445 + 0.895i)10-s + (0.696 + 0.717i)11-s + (0.213 − 0.976i)13-s + (0.881 + 0.473i)14-s + (−0.998 − 0.0615i)16-s + (0.932 + 0.361i)17-s + (−0.602 + 0.798i)19-s + (0.952 − 0.303i)20-s + (0.0307 − 0.999i)22-s + (−0.696 + 0.717i)23-s + ⋯
L(s)  = 1  + (−0.696 − 0.717i)2-s + (−0.0307 + 0.999i)4-s + (−0.332 − 0.943i)5-s + (−0.952 + 0.303i)7-s + (0.739 − 0.673i)8-s + (−0.445 + 0.895i)10-s + (0.696 + 0.717i)11-s + (0.213 − 0.976i)13-s + (0.881 + 0.473i)14-s + (−0.998 − 0.0615i)16-s + (0.932 + 0.361i)17-s + (−0.602 + 0.798i)19-s + (0.952 − 0.303i)20-s + (0.0307 − 0.999i)22-s + (−0.696 + 0.717i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.943 + 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.943 + 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $-0.943 + 0.331i$
Analytic conductor: \(99.6199\)
Root analytic conductor: \(99.6199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (1:\ ),\ -0.943 + 0.331i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05249399570 - 0.3081056051i\)
\(L(\frac12)\) \(\approx\) \(-0.05249399570 - 0.3081056051i\)
\(L(1)\) \(\approx\) \(0.5614540632 - 0.2325802577i\)
\(L(1)\) \(\approx\) \(0.5614540632 - 0.2325802577i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (-0.696 - 0.717i)T \)
5 \( 1 + (-0.332 - 0.943i)T \)
7 \( 1 + (-0.952 + 0.303i)T \)
11 \( 1 + (0.696 + 0.717i)T \)
13 \( 1 + (0.213 - 0.976i)T \)
17 \( 1 + (0.932 + 0.361i)T \)
19 \( 1 + (-0.602 + 0.798i)T \)
23 \( 1 + (-0.696 + 0.717i)T \)
29 \( 1 + (0.650 - 0.759i)T \)
31 \( 1 + (-0.552 + 0.833i)T \)
37 \( 1 + (-0.0922 - 0.995i)T \)
41 \( 1 + (0.650 + 0.759i)T \)
43 \( 1 + (-0.816 - 0.577i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (0.602 - 0.798i)T \)
59 \( 1 + (-0.952 - 0.303i)T \)
61 \( 1 + (-0.779 + 0.626i)T \)
67 \( 1 + (0.952 + 0.303i)T \)
71 \( 1 + (0.982 - 0.183i)T \)
73 \( 1 + (0.982 - 0.183i)T \)
79 \( 1 + (0.650 - 0.759i)T \)
83 \( 1 + (-0.952 + 0.303i)T \)
89 \( 1 + (0.850 - 0.526i)T \)
97 \( 1 + (-0.153 + 0.988i)T \)
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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

−22.28301839413271944533653127673, −21.50453268942692534557209039739, −20.08182918950842202997514502396, −19.53530350925201890280392679412, −18.71399698312361373169118987291, −18.47378489026446380636632313393, −17.14044411530497851817474100918, −16.57531412154302736118836140449, −15.89926216495898407376086762434, −15.06843875539929120014093121060, −14.07792595024122397138229610339, −13.84216051593891536593136785396, −12.3620115945885203139157673892, −11.3148444806946112707141930384, −10.67440320721573440782669619770, −9.76547561315045606502533334703, −9.078468377269905850612724776145, −8.10349464776505293541874253121, −7.109566281498286132514404098678, −6.52422358972447123417559258951, −5.96575082423213852036335506916, −4.48108046396141631812746543455, −3.51497559168849507988721599413, −2.41809316427125563328783198907, −0.97624411304687543273740849163, 0.1099194266217258583847993187, 1.151832894708735714184368001372, 2.13362976597880346231729475517, 3.49877733463891483591878534848, 3.93848429052695442605630341045, 5.27829116491842344873746096087, 6.32719613210012645794827300915, 7.5346228760412099550303646749, 8.23233907596447384952374038078, 9.07161578297946570828488827983, 9.83193575866922991998381223902, 10.42292554553215396544397197906, 11.73163135691154174096376084488, 12.40653321272719315792751977219, 12.7106241953602567836647686233, 13.72762629121941738806689507104, 15.07346147691598677683098660213, 15.926827156493463060311339987818, 16.60944335769330287615519310718, 17.28169341693234132240261490960, 18.12120427205808881048790230845, 19.06274197567198189540532990839, 19.83651136046271546490037608648, 20.0708256146359371185084351908, 21.17528195714444264327567440457

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