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Properties

Label 1-1007-1007.635-r1-0-0
Degree $1$
Conductor $1007$
Sign $-0.813 - 0.582i$
Analytic cond. $108.217$
Root an. cond. $108.217$
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  + (−0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)6-s + 7-s + 8-s + (−0.5 + 0.866i)9-s + (0.5 − 0.866i)10-s + 11-s + 12-s + (0.5 − 0.866i)13-s + (−0.5 − 0.866i)14-s + (0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)6-s + 7-s + 8-s + (−0.5 + 0.866i)9-s + (0.5 − 0.866i)10-s + 11-s + 12-s + (0.5 − 0.866i)13-s + (−0.5 − 0.866i)14-s + (0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1007 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1007 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1007\)    =    \(19 \cdot 53\)
Sign: $-0.813 - 0.582i$
Analytic conductor: \(108.217\)
Root analytic conductor: \(108.217\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1007} (635, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1007,\ (1:\ ),\ -0.813 - 0.582i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4888371887 - 1.522147210i\)
\(L(\frac12)\) \(\approx\) \(0.4888371887 - 1.522147210i\)
\(L(1)\) \(\approx\) \(0.7406634328 - 0.5241197023i\)
\(L(1)\) \(\approx\) \(0.7406634328 - 0.5241197023i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
53 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + T \)
37 \( 1 - T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 - T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−21.56992462388942673805990882992, −21.24700679687086360072547852375, −20.15309785191619708136802927239, −19.47898245410588224487688447013, −18.21930586736254046293945206384, −17.47498972725250576075997517825, −17.09198171646247349433551023879, −16.41795507931256080808382332691, −15.6217140675846004881860889705, −14.789935144665835854494456869195, −14.14163840855838684375998629595, −13.28124155540152183362305234737, −11.93285171228379401598620278637, −11.26043176526523967457610842112, −10.31367768815962449842970702450, −9.482220115092494000285630046915, −8.74512061598750195471561652222, −8.34058376452012180504519769823, −6.803306303189873852125427186588, −6.165934254151938631241404926428, −5.18039899037823487756781801661, −4.63661874754000024627236044087, −3.82910174167180878433254578154, −1.67034453907141768406853183552, −1.09483229126476550912847466710, 0.490933530995236546775608814402, 1.40722480071000346986248051944, 2.2595123045354452470852564895, 3.084269690852655192777663463090, 4.40982081062296958662823506978, 5.41990413579486945169373685985, 6.562000947950729211742338293926, 7.240322179250336258169444908026, 8.209940237574641588657284870203, 8.90504685650909591552995601275, 10.18947602104747156372418015133, 10.76562804694215861932722993473, 11.57545399142268928189923605911, 11.989010475114286595331728141242, 13.15818261504869010566017078276, 13.81843603317493604736774280396, 14.4386066720449420496970384786, 15.70127158217521113965069977358, 17.076683571146475537311451011838, 17.44310659112755095183327142211, 18.043510081441210773328747025580, 18.71806749194113373202101401446, 19.3430305574969558838498533031, 20.36173086083578570833524951635, 20.988365194493379010148953108413

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