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Properties

Label 1-407-407.105-r0-0-0
Degree $1$
Conductor $407$
Sign $-0.629 - 0.777i$
Analytic cond. $1.89010$
Root an. cond. $1.89010$
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  + (0.587 + 0.809i)2-s + (−0.309 − 0.951i)3-s + (−0.309 + 0.951i)4-s + (−0.587 + 0.809i)5-s + (0.587 − 0.809i)6-s + (−0.309 + 0.951i)7-s + (−0.951 + 0.309i)8-s + (−0.809 + 0.587i)9-s − 10-s + 12-s + (−0.587 − 0.809i)13-s + (−0.951 + 0.309i)14-s + (0.951 + 0.309i)15-s + (−0.809 − 0.587i)16-s + (0.587 − 0.809i)17-s + (−0.951 − 0.309i)18-s + ⋯
L(s)  = 1  + (0.587 + 0.809i)2-s + (−0.309 − 0.951i)3-s + (−0.309 + 0.951i)4-s + (−0.587 + 0.809i)5-s + (0.587 − 0.809i)6-s + (−0.309 + 0.951i)7-s + (−0.951 + 0.309i)8-s + (−0.809 + 0.587i)9-s − 10-s + 12-s + (−0.587 − 0.809i)13-s + (−0.951 + 0.309i)14-s + (0.951 + 0.309i)15-s + (−0.809 − 0.587i)16-s + (0.587 − 0.809i)17-s + (−0.951 − 0.309i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.629 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.629 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $-0.629 - 0.777i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (105, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ -0.629 - 0.777i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05905951732 + 0.1238463045i\)
\(L(\frac12)\) \(\approx\) \(-0.05905951732 + 0.1238463045i\)
\(L(1)\) \(\approx\) \(0.6785520271 + 0.3183182399i\)
\(L(1)\) \(\approx\) \(0.6785520271 + 0.3183182399i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.587 + 0.809i)T \)
3 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (-0.587 + 0.809i)T \)
7 \( 1 + (-0.309 + 0.951i)T \)
13 \( 1 + (-0.587 - 0.809i)T \)
17 \( 1 + (0.587 - 0.809i)T \)
19 \( 1 + (-0.951 + 0.309i)T \)
23 \( 1 - iT \)
29 \( 1 + (-0.951 - 0.309i)T \)
31 \( 1 + (-0.587 - 0.809i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + iT \)
47 \( 1 + (0.309 + 0.951i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (-0.951 - 0.309i)T \)
61 \( 1 + (-0.587 + 0.809i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (-0.587 - 0.809i)T \)
83 \( 1 + (0.809 + 0.587i)T \)
89 \( 1 + iT \)
97 \( 1 + (0.587 + 0.809i)T \)
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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

−23.63907970821935891631020184299, −22.9138410733098721601628328289, −21.82735908845031051170548521351, −21.25292931000444403023266954125, −20.362815835120939849457663471, −19.72843762994381288901733785052, −19.01750759106205359291205383757, −17.33322578899330726088848772606, −16.76862712855610242129597732947, −15.772701177186178102688008356510, −14.899475781476795843012651432241, −14.00871033404593630637821235393, −12.89695959857695598530985237178, −12.125909161127415932901667961048, −11.19009734676101706526205469987, −10.44470299835954004362072895369, −9.51153872510112646310562048599, −8.72643703298634879893483185548, −7.15234301219014681906638688226, −5.78588406244905513258893844877, −4.82979492951647738740665393882, −4.02169665897122054407411294977, −3.44905110541799187019519757419, −1.64005738479378618885601592667, −0.06554591772820765147212976593, 2.45044232134687457587875269117, 3.11770680669193519660457031475, 4.658644658159650777808649795968, 5.8638578014430192108815454097, 6.39117999993028991825012326569, 7.567998722509953558136569414083, 7.97660572295667442004131646754, 9.268691934882779486990272961804, 10.853764402900237426692567578423, 11.90569056622191861652255393939, 12.47405300878300028766821238648, 13.28985331992003799617704936783, 14.59943036541351833376823149394, 14.90507134531914496027679958608, 16.076601884062934570690436665973, 16.91062776198505258902136321071, 18.03996905705000277478085201517, 18.5581041313492045287929055488, 19.369458177948989212386838256754, 20.635785473634778712176617602822, 21.96140904373562131183490938022, 22.567731316245460438410427815095, 23.06184131217586283804385062504, 24.02914383843199270965048527485, 24.845737587579895180477698821797

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