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Properties

Label 1-469-469.107-r0-0-0
Degree $1$
Conductor $469$
Sign $0.137 - 0.990i$
Analytic cond. $2.17802$
Root an. cond. $2.17802$
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  + (0.928 − 0.371i)2-s + (0.981 − 0.189i)3-s + (0.723 − 0.690i)4-s + (0.0475 − 0.998i)5-s + (0.841 − 0.540i)6-s + (0.415 − 0.909i)8-s + (0.928 − 0.371i)9-s + (−0.327 − 0.945i)10-s + (−0.888 + 0.458i)11-s + (0.580 − 0.814i)12-s + (0.415 + 0.909i)13-s + (−0.142 − 0.989i)15-s + (0.0475 − 0.998i)16-s + (0.235 − 0.971i)17-s + (0.723 − 0.690i)18-s + (−0.786 + 0.618i)19-s + ⋯
L(s)  = 1  + (0.928 − 0.371i)2-s + (0.981 − 0.189i)3-s + (0.723 − 0.690i)4-s + (0.0475 − 0.998i)5-s + (0.841 − 0.540i)6-s + (0.415 − 0.909i)8-s + (0.928 − 0.371i)9-s + (−0.327 − 0.945i)10-s + (−0.888 + 0.458i)11-s + (0.580 − 0.814i)12-s + (0.415 + 0.909i)13-s + (−0.142 − 0.989i)15-s + (0.0475 − 0.998i)16-s + (0.235 − 0.971i)17-s + (0.723 − 0.690i)18-s + (−0.786 + 0.618i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.137 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.137 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(469\)    =    \(7 \cdot 67\)
Sign: $0.137 - 0.990i$
Analytic conductor: \(2.17802\)
Root analytic conductor: \(2.17802\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{469} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 469,\ (0:\ ),\ 0.137 - 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.456362449 - 2.138305510i\)
\(L(\frac12)\) \(\approx\) \(2.456362449 - 2.138305510i\)
\(L(1)\) \(\approx\) \(2.105779089 - 1.072642943i\)
\(L(1)\) \(\approx\) \(2.105779089 - 1.072642943i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
67 \( 1 \)
good2 \( 1 + (0.928 - 0.371i)T \)
3 \( 1 + (0.981 - 0.189i)T \)
5 \( 1 + (0.0475 - 0.998i)T \)
11 \( 1 + (-0.888 + 0.458i)T \)
13 \( 1 + (0.415 + 0.909i)T \)
17 \( 1 + (0.235 - 0.971i)T \)
19 \( 1 + (-0.786 + 0.618i)T \)
23 \( 1 + (-0.327 + 0.945i)T \)
29 \( 1 + T \)
31 \( 1 + (0.580 + 0.814i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.959 + 0.281i)T \)
43 \( 1 + (-0.959 + 0.281i)T \)
47 \( 1 + (-0.327 + 0.945i)T \)
53 \( 1 + (0.723 - 0.690i)T \)
59 \( 1 + (0.580 + 0.814i)T \)
61 \( 1 + (-0.888 - 0.458i)T \)
71 \( 1 + (-0.959 - 0.281i)T \)
73 \( 1 + (0.0475 + 0.998i)T \)
79 \( 1 + (0.580 - 0.814i)T \)
83 \( 1 + (0.841 + 0.540i)T \)
89 \( 1 + (0.981 + 0.189i)T \)
97 \( 1 + T \)
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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

−23.967037462123813227856251931792, −23.27410076767951008092927696838, −22.233222553005473808792441198789, −21.58153711508012710550142346942, −20.87301401806614074905409208412, −19.9803255721127539987560444594, −19.03968298102097194415110861689, −18.17556318507522309210027170038, −17.00954019094281826987719137736, −15.79524624788221623853211795058, −15.21305369656070266489635730138, −14.67250178326986535923343481619, −13.54843696336301573668773238080, −13.23144111130469979310012531570, −12.0050910522450007095735758651, −10.54397515325535801526174627804, −10.39663896424470708198666264126, −8.43284639708149109918807148566, −8.05236916609394781035040400905, −6.87719903833106561302717078226, −6.05115651818161391328660470540, −4.80032948404417602749427665723, −3.65500763651395234653959266223, −2.94242375510022557963643867925, −2.1259010314076881815590123960, 1.362025943482415640235335411970, 2.20140442635720413302004198729, 3.37407777278528449094524155279, 4.395793208889813511901874068611, 5.14280436684511178050959911802, 6.4564858354916205958907362084, 7.5191430323465909214294364890, 8.522730172632845557647020445303, 9.55843322023128844684809665078, 10.33542299273506802886471042993, 11.79866262721774627936713743014, 12.42879182892209405646632586323, 13.38405345152733370171363722363, 13.81984339523520672222527491054, 14.82771389538385095777477696747, 15.83736056559257362333892212619, 16.28319335303167627146492873060, 17.85221462712379885271135700363, 18.905435859614555573703651225595, 19.66215606512630356271469107652, 20.43651715668381743776240263058, 21.18906438293173672483044453351, 21.44286497998290978162943497386, 23.13683291313075832198675957329, 23.57841395797586258023390856431

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