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Properties

Label 1-1037-1037.767-r1-0-0
Degree $1$
Conductor $1037$
Sign $-0.475 - 0.879i$
Analytic cond. $111.441$
Root an. cond. $111.441$
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  + (−0.913 + 0.406i)2-s + (0.987 − 0.156i)3-s + (0.669 − 0.743i)4-s + (−0.838 + 0.544i)5-s + (−0.838 + 0.544i)6-s + (−0.777 − 0.629i)7-s + (−0.309 + 0.951i)8-s + (0.951 − 0.309i)9-s + (0.544 − 0.838i)10-s + (0.707 − 0.707i)11-s + (0.544 − 0.838i)12-s + (0.5 − 0.866i)13-s + (0.965 + 0.258i)14-s + (−0.743 + 0.669i)15-s + (−0.104 − 0.994i)16-s + ⋯
L(s)  = 1  + (−0.913 + 0.406i)2-s + (0.987 − 0.156i)3-s + (0.669 − 0.743i)4-s + (−0.838 + 0.544i)5-s + (−0.838 + 0.544i)6-s + (−0.777 − 0.629i)7-s + (−0.309 + 0.951i)8-s + (0.951 − 0.309i)9-s + (0.544 − 0.838i)10-s + (0.707 − 0.707i)11-s + (0.544 − 0.838i)12-s + (0.5 − 0.866i)13-s + (0.965 + 0.258i)14-s + (−0.743 + 0.669i)15-s + (−0.104 − 0.994i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1037 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.475 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1037 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.475 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1037\)    =    \(17 \cdot 61\)
Sign: $-0.475 - 0.879i$
Analytic conductor: \(111.441\)
Root analytic conductor: \(111.441\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1037} (767, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1037,\ (1:\ ),\ -0.475 - 0.879i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6054128456 - 1.016043622i\)
\(L(\frac12)\) \(\approx\) \(0.6054128456 - 1.016043622i\)
\(L(1)\) \(\approx\) \(0.8343644856 - 0.1176324196i\)
\(L(1)\) \(\approx\) \(0.8343644856 - 0.1176324196i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
61 \( 1 \)
good2 \( 1 + (-0.913 + 0.406i)T \)
3 \( 1 + (0.987 - 0.156i)T \)
5 \( 1 + (-0.838 + 0.544i)T \)
7 \( 1 + (-0.777 - 0.629i)T \)
11 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (0.994 + 0.104i)T \)
23 \( 1 + (-0.891 - 0.453i)T \)
29 \( 1 + (0.258 - 0.965i)T \)
31 \( 1 + (-0.358 - 0.933i)T \)
37 \( 1 + (0.987 + 0.156i)T \)
41 \( 1 + (-0.987 - 0.156i)T \)
43 \( 1 + (-0.669 - 0.743i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (-0.913 + 0.406i)T \)
67 \( 1 + (-0.207 + 0.978i)T \)
71 \( 1 + (0.838 + 0.544i)T \)
73 \( 1 + (0.544 - 0.838i)T \)
79 \( 1 + (-0.998 - 0.0523i)T \)
83 \( 1 + (0.406 + 0.913i)T \)
89 \( 1 + (0.587 - 0.809i)T \)
97 \( 1 + (0.933 - 0.358i)T \)
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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

−21.5412449704277315581600276038, −20.40287177675852921277212135374, −19.92539726503108125472346261899, −19.54573668914402604373565161936, −18.607714526313540382018508021547, −18.10727688679152766019083026188, −16.71399279728706539220517884670, −16.125974927024113125889898690397, −15.62215826422150159274320867673, −14.74955285363272097966347755983, −13.62116229145773410999118245248, −12.64841180569116145572196901260, −12.07700598904043255099024148402, −11.33301807133950503649621204728, −10.05032407737623356717547366872, −9.36659843747198105394234608352, −8.89607352855105671642963522520, −8.11215336506812244045346772933, −7.2213697968584338105458364220, −6.516814293008042030655168980112, −4.83862599656184387341306686393, −3.70077332294855957697012485055, −3.30321976774134175379282720904, −2.01558278311328070417604738893, −1.22518770676921101588975381564, 0.33011300523919942502233634199, 1.15140058858311947163470688899, 2.62775200635653469947806791857, 3.37648273199419651254470142662, 4.17888798762987321646262409402, 5.957306231911184694746690171958, 6.62670695643289284865679270312, 7.58184086016148187085891038242, 7.98377149804682169170589087678, 8.88827890695178694688489165689, 9.78806280885697215765550744407, 10.41451485673666598965440637403, 11.36167542028933340539342735436, 12.261131218928125799703261667045, 13.528481297572193092283930309978, 14.09785859772669377681801429696, 15.01503384832783800854052419370, 15.68809283358406420762245120653, 16.24862125684911723138791568932, 17.165454541815122712400897198396, 18.38975443083715930468188567495, 18.708000942180402801585163605705, 19.54621236376658405456966407514, 20.13953508347694501317120081056, 20.487863426221490502514907983067

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