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Properties

Label 1-633-633.266-r1-0-0
Degree $1$
Conductor $633$
Sign $0.995 + 0.0925i$
Analytic cond. $68.0252$
Root an. cond. $68.0252$
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  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (−0.309 + 0.951i)5-s + (−0.809 − 0.587i)7-s + (−0.309 − 0.951i)8-s + (0.309 + 0.951i)10-s + (−0.309 + 0.951i)11-s + (−0.809 − 0.587i)13-s − 14-s + (−0.809 − 0.587i)16-s + (0.809 + 0.587i)17-s + (−0.809 − 0.587i)19-s + (0.809 + 0.587i)20-s + (0.309 + 0.951i)22-s + (0.809 − 0.587i)23-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (−0.309 + 0.951i)5-s + (−0.809 − 0.587i)7-s + (−0.309 − 0.951i)8-s + (0.309 + 0.951i)10-s + (−0.309 + 0.951i)11-s + (−0.809 − 0.587i)13-s − 14-s + (−0.809 − 0.587i)16-s + (0.809 + 0.587i)17-s + (−0.809 − 0.587i)19-s + (0.809 + 0.587i)20-s + (0.309 + 0.951i)22-s + (0.809 − 0.587i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.995 + 0.0925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.995 + 0.0925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(633\)    =    \(3 \cdot 211\)
Sign: $0.995 + 0.0925i$
Analytic conductor: \(68.0252\)
Root analytic conductor: \(68.0252\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{633} (266, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 633,\ (1:\ ),\ 0.995 + 0.0925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.201432834 + 0.1020656927i\)
\(L(\frac12)\) \(\approx\) \(2.201432834 + 0.1020656927i\)
\(L(1)\) \(\approx\) \(1.322640783 - 0.2978506738i\)
\(L(1)\) \(\approx\) \(1.322640783 - 0.2978506738i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
211 \( 1 \)
good2 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (-0.309 + 0.951i)T \)
7 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (-0.309 + 0.951i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
17 \( 1 + (0.809 + 0.587i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (0.809 - 0.587i)T \)
29 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 + T \)
37 \( 1 + (0.309 + 0.951i)T \)
41 \( 1 + (-0.309 + 0.951i)T \)
43 \( 1 + T \)
47 \( 1 + (0.809 + 0.587i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (-0.309 + 0.951i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (-0.309 + 0.951i)T \)
89 \( 1 + (0.809 + 0.587i)T \)
97 \( 1 + (0.309 + 0.951i)T \)
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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

−23.04316366029275008361381375233, −21.763405958287660116748365048398, −21.36038352164528002171610877269, −20.529000874599001332759104453459, −19.34277270269743254355137850378, −18.85446877198777914774748796317, −17.27395601349180229648385653287, −16.767124760974629359692932919103, −15.92431543781388613015466676191, −15.47474805525906547067075751563, −14.25754306945971192281338366256, −13.55190726022479638077265130505, −12.51037504345915010358902067645, −12.21217022409255962818558130461, −11.19675828646222282121555970819, −9.707967455608927351784965992993, −8.82026082036331004911704332806, −8.04566603151733909835598493642, −7.04652735624271436429560816759, −5.94623720052923610645579408898, −5.31514667885939723233672297652, −4.30952502374721578602430903440, −3.31778383963873368187329862371, −2.31923011592207555484590634684, −0.493492701125712325457179243346, 0.871502926408101587559448135472, 2.50285551456299802013962196473, 3.024459646498674291283008772259, 4.12900311893583176008975309772, 4.96918168833968415634444171699, 6.33328181469243642989145428458, 6.87868273599720119692438319773, 7.85343774961902106451152474097, 9.5262901132650430243038223678, 10.370580309915656068768779257364, 10.64431663018691259308148020891, 11.96931217655084839584276297968, 12.638680649496800031915216653982, 13.38600117801268948986952684097, 14.4836889061425490728892910373, 15.019007688483729322234191095660, 15.736822547777728607897245254821, 16.95078152212081713776422825993, 17.93584703913836458760811771301, 19.05584606988680764546654003038, 19.4605864140233312342233760265, 20.25720218022506228517070818729, 21.19184430869468222239897745632, 22.10625595986566015327090806449, 22.753485746552923814922303765439

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