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Properties

Label 1-1337-1337.130-r0-0-0
Degree $1$
Conductor $1337$
Sign $0.536 - 0.843i$
Analytic cond. $6.20900$
Root an. cond. $6.20900$
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  + (0.970 + 0.240i)2-s + (−0.917 + 0.396i)3-s + (0.884 + 0.466i)4-s + (−0.962 − 0.272i)5-s + (−0.986 + 0.164i)6-s + (0.746 + 0.665i)8-s + (0.685 − 0.728i)9-s + (−0.868 − 0.495i)10-s + (0.451 − 0.892i)11-s + (−0.997 − 0.0770i)12-s + (−0.909 + 0.416i)13-s + (0.991 − 0.131i)15-s + (0.565 + 0.824i)16-s + (0.874 − 0.485i)17-s + (0.840 − 0.542i)18-s + (−0.868 + 0.495i)19-s + ⋯
L(s)  = 1  + (0.970 + 0.240i)2-s + (−0.917 + 0.396i)3-s + (0.884 + 0.466i)4-s + (−0.962 − 0.272i)5-s + (−0.986 + 0.164i)6-s + (0.746 + 0.665i)8-s + (0.685 − 0.728i)9-s + (−0.868 − 0.495i)10-s + (0.451 − 0.892i)11-s + (−0.997 − 0.0770i)12-s + (−0.909 + 0.416i)13-s + (0.991 − 0.131i)15-s + (0.565 + 0.824i)16-s + (0.874 − 0.485i)17-s + (0.840 − 0.542i)18-s + (−0.868 + 0.495i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1337 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.536 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1337 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.536 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1337\)    =    \(7 \cdot 191\)
Sign: $0.536 - 0.843i$
Analytic conductor: \(6.20900\)
Root analytic conductor: \(6.20900\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1337} (130, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1337,\ (0:\ ),\ 0.536 - 0.843i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.080016727 - 0.5930019653i\)
\(L(\frac12)\) \(\approx\) \(1.080016727 - 0.5930019653i\)
\(L(1)\) \(\approx\) \(1.124170104 + 0.07723927103i\)
\(L(1)\) \(\approx\) \(1.124170104 + 0.07723927103i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
191 \( 1 \)
good2 \( 1 + (0.970 + 0.240i)T \)
3 \( 1 + (-0.917 + 0.396i)T \)
5 \( 1 + (-0.962 - 0.272i)T \)
11 \( 1 + (0.451 - 0.892i)T \)
13 \( 1 + (-0.909 + 0.416i)T \)
17 \( 1 + (0.874 - 0.485i)T \)
19 \( 1 + (-0.868 + 0.495i)T \)
23 \( 1 + (-0.739 - 0.673i)T \)
29 \( 1 + (0.180 - 0.983i)T \)
31 \( 1 + (-0.821 + 0.569i)T \)
37 \( 1 + (-0.821 - 0.569i)T \)
41 \( 1 + (0.789 - 0.614i)T \)
43 \( 1 + (-0.461 - 0.887i)T \)
47 \( 1 + (-0.360 - 0.932i)T \)
53 \( 1 + (-0.889 + 0.456i)T \)
59 \( 1 + (0.999 + 0.0220i)T \)
61 \( 1 + (-0.421 - 0.906i)T \)
67 \( 1 + (0.391 - 0.920i)T \)
71 \( 1 + (0.828 + 0.560i)T \)
73 \( 1 + (-0.256 - 0.966i)T \)
79 \( 1 + (-0.942 - 0.335i)T \)
83 \( 1 + (-0.213 - 0.976i)T \)
89 \( 1 + (0.930 + 0.366i)T \)
97 \( 1 + (0.922 + 0.386i)T \)
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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

−21.28573334686313437429100361316, −20.14904908765521652589199190785, −19.57232886940034978660571540296, −19.03163076609413985232015024106, −17.986053145631509901494774799778, −17.140158660400116540476767279106, −16.36976515321245960400608097934, −15.59213591779885399920220188809, −14.82618048565757139518277141057, −14.30638629631894221684003546480, −12.81755338272497202175761345127, −12.68700310615916351365666188069, −11.823222709849044423077899662189, −11.29311071420819638426471047985, −10.42191954348156304868548255896, −9.75194944760179602349742710014, −8.043373696556976717254391885193, −7.32716430762494179921537537040, −6.74975059012364411410257965978, −5.82992727666115044156286661101, −4.89413938765350284194068076775, −4.30014437448221042558616070722, −3.34054573883550989090049652865, −2.20034734064763244684493675361, −1.204435699286633309194554062204, 0.4047397323909755639968299498, 1.92998488520209971445614036276, 3.37885281506349805042542477962, 3.95551101092188150540281596698, 4.74341129369279996935970575207, 5.49552494028437168106961006140, 6.3459071510833940308590984257, 7.13391012241168737385152354718, 7.968759651451588340969978018156, 8.96896951557939110725000127221, 10.22030208627465225631971485868, 10.95703461275252764020462413132, 11.82516223220043207753974589433, 12.13617121473792708330247482514, 12.83469955142014396621697461064, 14.12407088792009762925255396436, 14.63865378314930051468425941554, 15.577098135977702930007547157052, 16.20319720967111670558809848151, 16.73557452668179698858956186632, 17.29377779362378715116839021534, 18.67056351591227793918920464839, 19.32673215427387805539689841101, 20.24419629485044738879854003263, 21.05777445166200166903570544845

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