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Properties

Label 1-735-735.269-r0-0-0
Degree $1$
Conductor $735$
Sign $-0.710 - 0.703i$
Analytic cond. $3.41332$
Root an. cond. $3.41332$
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  + (0.826 − 0.563i)2-s + (0.365 − 0.930i)4-s + (−0.222 − 0.974i)8-s + (−0.0747 − 0.997i)11-s + (−0.900 − 0.433i)13-s + (−0.733 − 0.680i)16-s + (0.988 + 0.149i)17-s + (0.5 − 0.866i)19-s + (−0.623 − 0.781i)22-s + (−0.988 + 0.149i)23-s + (−0.988 + 0.149i)26-s + (−0.623 + 0.781i)29-s + (0.5 + 0.866i)31-s + (−0.988 − 0.149i)32-s + (0.900 − 0.433i)34-s + ⋯
L(s)  = 1  + (0.826 − 0.563i)2-s + (0.365 − 0.930i)4-s + (−0.222 − 0.974i)8-s + (−0.0747 − 0.997i)11-s + (−0.900 − 0.433i)13-s + (−0.733 − 0.680i)16-s + (0.988 + 0.149i)17-s + (0.5 − 0.866i)19-s + (−0.623 − 0.781i)22-s + (−0.988 + 0.149i)23-s + (−0.988 + 0.149i)26-s + (−0.623 + 0.781i)29-s + (0.5 + 0.866i)31-s + (−0.988 − 0.149i)32-s + (0.900 − 0.433i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.710 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.710 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.710 - 0.703i$
Analytic conductor: \(3.41332\)
Root analytic conductor: \(3.41332\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 735,\ (0:\ ),\ -0.710 - 0.703i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7427252558 - 1.806733871i\)
\(L(\frac12)\) \(\approx\) \(0.7427252558 - 1.806733871i\)
\(L(1)\) \(\approx\) \(1.254862252 - 0.8672566968i\)
\(L(1)\) \(\approx\) \(1.254862252 - 0.8672566968i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.826 - 0.563i)T \)
11 \( 1 + (-0.0747 - 0.997i)T \)
13 \( 1 + (-0.900 - 0.433i)T \)
17 \( 1 + (0.988 + 0.149i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.988 + 0.149i)T \)
29 \( 1 + (-0.623 + 0.781i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (-0.365 - 0.930i)T \)
41 \( 1 + (-0.222 - 0.974i)T \)
43 \( 1 + (0.222 - 0.974i)T \)
47 \( 1 + (-0.826 + 0.563i)T \)
53 \( 1 + (0.365 - 0.930i)T \)
59 \( 1 + (0.955 - 0.294i)T \)
61 \( 1 + (-0.365 - 0.930i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.623 - 0.781i)T \)
73 \( 1 + (0.826 + 0.563i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.900 - 0.433i)T \)
89 \( 1 + (0.0747 - 0.997i)T \)
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

−22.784520274267534608835777396703, −22.25058897575658184929277727257, −21.24212826999021720481373858316, −20.61732037462186085911228258679, −19.80678056354630952623962621121, −18.65600941064181986947010293213, −17.76714684122352593751514149490, −16.85431627414691493661570276517, −16.34235358987527635934661463651, −15.21730928525062186962108030285, −14.6885533917991215952378344645, −13.8949437555746391102985576216, −12.985572948276345962714056871171, −12.01976165840968316357224434579, −11.73732938900735479685162105714, −10.159265911677431443768536165857, −9.52054986759160798008296721315, −8.030657942978514987392441703701, −7.59189973343609386284399875916, −6.569560740270436504851719035189, −5.65831487366902495748217087339, −4.737770768181107097643564837875, −3.951909595021983972445257703330, −2.80415294686316550376174347932, −1.79204673090889560726497188930, 0.6699672443329109855379682849, 2.00248196858698264481282117314, 3.08053682231203806719954022415, 3.76363762161566585534043538870, 5.11723917564368445296835623086, 5.56253675455282584285761111848, 6.72219332218097947842042353566, 7.67685100928363795158187038748, 8.89716580001264193915600766219, 9.918231781907384452572302935493, 10.622822626567923695868191466947, 11.55012930326358334183189384910, 12.28714701666007535231042510956, 13.067639856569542943876732072713, 14.05871916966960203769813169992, 14.48762374297190231746498242259, 15.629544474592635506651873214992, 16.21934268531158746863079083079, 17.36307188201168419478713748010, 18.360550705612688794849841518607, 19.25238871345761135173107946272, 19.79685833929059729256089064954, 20.698723623861230254583665190203, 21.52989319567318462561707796328, 22.1048220965702206097600597752

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