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Properties

Label 1-828-828.583-r1-0-0
Degree $1$
Conductor $828$
Sign $0.926 + 0.376i$
Analytic cond. $88.9809$
Root an. cond. $88.9809$
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  + (−0.786 − 0.618i)5-s + (−0.580 + 0.814i)7-s + (−0.723 − 0.690i)11-s + (0.580 + 0.814i)13-s + (0.841 − 0.540i)17-s + (−0.841 − 0.540i)19-s + (0.235 + 0.971i)25-s + (−0.888 − 0.458i)29-s + (−0.981 + 0.189i)31-s + (0.959 − 0.281i)35-s + (−0.142 − 0.989i)37-s + (−0.786 − 0.618i)41-s + (−0.981 − 0.189i)43-s + (0.5 + 0.866i)47-s + (−0.327 − 0.945i)49-s + ⋯
L(s)  = 1  + (−0.786 − 0.618i)5-s + (−0.580 + 0.814i)7-s + (−0.723 − 0.690i)11-s + (0.580 + 0.814i)13-s + (0.841 − 0.540i)17-s + (−0.841 − 0.540i)19-s + (0.235 + 0.971i)25-s + (−0.888 − 0.458i)29-s + (−0.981 + 0.189i)31-s + (0.959 − 0.281i)35-s + (−0.142 − 0.989i)37-s + (−0.786 − 0.618i)41-s + (−0.981 − 0.189i)43-s + (0.5 + 0.866i)47-s + (−0.327 − 0.945i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.926 + 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.926 + 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(828\)    =    \(2^{2} \cdot 3^{2} \cdot 23\)
Sign: $0.926 + 0.376i$
Analytic conductor: \(88.9809\)
Root analytic conductor: \(88.9809\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{828} (583, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 828,\ (1:\ ),\ 0.926 + 0.376i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9482476002 + 0.1852408342i\)
\(L(\frac12)\) \(\approx\) \(0.9482476002 + 0.1852408342i\)
\(L(1)\) \(\approx\) \(0.7691783097 + 0.02653924077i\)
\(L(1)\) \(\approx\) \(0.7691783097 + 0.02653924077i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
23 \( 1 \)
good5 \( 1 + (-0.786 - 0.618i)T \)
7 \( 1 + (-0.580 + 0.814i)T \)
11 \( 1 + (-0.723 - 0.690i)T \)
13 \( 1 + (0.580 + 0.814i)T \)
17 \( 1 + (0.841 - 0.540i)T \)
19 \( 1 + (-0.841 - 0.540i)T \)
29 \( 1 + (-0.888 - 0.458i)T \)
31 \( 1 + (-0.981 + 0.189i)T \)
37 \( 1 + (-0.142 - 0.989i)T \)
41 \( 1 + (-0.786 - 0.618i)T \)
43 \( 1 + (-0.981 - 0.189i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.415 + 0.909i)T \)
59 \( 1 + (-0.580 - 0.814i)T \)
61 \( 1 + (-0.327 + 0.945i)T \)
67 \( 1 + (-0.723 + 0.690i)T \)
71 \( 1 + (0.959 + 0.281i)T \)
73 \( 1 + (0.841 + 0.540i)T \)
79 \( 1 + (0.995 - 0.0950i)T \)
83 \( 1 + (0.786 - 0.618i)T \)
89 \( 1 + (-0.654 + 0.755i)T \)
97 \( 1 + (0.928 - 0.371i)T \)
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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

−22.11320065749533974070373492577, −20.98256010439089326707875027124, −20.225105317187547748732723495991, −19.63314969954109483417972074911, −18.62057941600361586236858323556, −18.19569252878793235765267390420, −16.9059470121026529321908350154, −16.40775823243060356121491944171, −15.173698260598114595477323732173, −15.017584093496808655034469719975, −13.753560213259410480621702626959, −12.9000559108045827873300931204, −12.2749270994611694303392979569, −11.046276839235513567951241663807, −10.44979019786092469245601645095, −9.834960946431753298302560339506, −8.36003466107916073331426253552, −7.730725653138908533698060243869, −6.94064294741030108709075322298, −6.026253577233707471538411708088, −4.85142090789095513783736191377, −3.66367181814218341597381827471, −3.28300468689906456467243500251, −1.82738797177626585919969037536, −0.383611296021801691320563210286, 0.558361285290231987329392539010, 1.996286158888477659248434683022, 3.17056800028953856557362721717, 3.98632148553784381989531105055, 5.159246418133203795711206386286, 5.86599677959350766452240593467, 6.997313597718095215288499487855, 7.9640017973839502726233419915, 8.841350964465030538939406527697, 9.33046732252010115698387396660, 10.67192399987049029153393994814, 11.45902323866669810614704229561, 12.25338759866461652141377217933, 12.97494467630186812606773799169, 13.787211158500322546896749820475, 14.96058676616062185937390133307, 15.699976594560624919196020718006, 16.31738065197971576256900193076, 16.92617986878403171503425848253, 18.35110043793847186014569667474, 18.84875319916017551784947470289, 19.468481870240124125121642492845, 20.50382052193626673820234757215, 21.21681982122013875079827573393, 21.90511321637519044995893977914

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