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Properties

Label 1-1007-1007.501-r0-0-0
Degree $1$
Conductor $1007$
Sign $-0.938 + 0.346i$
Analytic cond. $4.67648$
Root an. cond. $4.67648$
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  + (−0.200 − 0.979i)2-s + (0.692 − 0.721i)3-s + (−0.919 + 0.391i)4-s + (−0.845 + 0.534i)5-s + (−0.845 − 0.534i)6-s + (−0.748 + 0.663i)7-s + (0.568 + 0.822i)8-s + (−0.0402 − 0.999i)9-s + (0.692 + 0.721i)10-s + (−0.354 + 0.935i)11-s + (−0.354 + 0.935i)12-s + (0.799 − 0.600i)13-s + (0.799 + 0.600i)14-s + (−0.200 + 0.979i)15-s + (0.692 − 0.721i)16-s + (0.428 + 0.903i)17-s + ⋯
L(s)  = 1  + (−0.200 − 0.979i)2-s + (0.692 − 0.721i)3-s + (−0.919 + 0.391i)4-s + (−0.845 + 0.534i)5-s + (−0.845 − 0.534i)6-s + (−0.748 + 0.663i)7-s + (0.568 + 0.822i)8-s + (−0.0402 − 0.999i)9-s + (0.692 + 0.721i)10-s + (−0.354 + 0.935i)11-s + (−0.354 + 0.935i)12-s + (0.799 − 0.600i)13-s + (0.799 + 0.600i)14-s + (−0.200 + 0.979i)15-s + (0.692 − 0.721i)16-s + (0.428 + 0.903i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1007 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.938 + 0.346i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1007 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.938 + 0.346i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1007\)    =    \(19 \cdot 53\)
Sign: $-0.938 + 0.346i$
Analytic conductor: \(4.67648\)
Root analytic conductor: \(4.67648\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1007} (501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1007,\ (0:\ ),\ -0.938 + 0.346i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.07999122186 - 0.4477487418i\)
\(L(\frac12)\) \(\approx\) \(-0.07999122186 - 0.4477487418i\)
\(L(1)\) \(\approx\) \(0.6100804916 - 0.4088348999i\)
\(L(1)\) \(\approx\) \(0.6100804916 - 0.4088348999i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
53 \( 1 \)
good2 \( 1 + (-0.200 - 0.979i)T \)
3 \( 1 + (0.692 - 0.721i)T \)
5 \( 1 + (-0.845 + 0.534i)T \)
7 \( 1 + (-0.748 + 0.663i)T \)
11 \( 1 + (-0.354 + 0.935i)T \)
13 \( 1 + (0.799 - 0.600i)T \)
17 \( 1 + (0.428 + 0.903i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.632 + 0.774i)T \)
31 \( 1 + (-0.354 - 0.935i)T \)
37 \( 1 + (-0.970 - 0.239i)T \)
41 \( 1 + (-0.632 - 0.774i)T \)
43 \( 1 + (0.278 - 0.960i)T \)
47 \( 1 + (-0.845 - 0.534i)T \)
59 \( 1 + (-0.0402 + 0.999i)T \)
61 \( 1 + (0.428 - 0.903i)T \)
67 \( 1 + (-0.919 + 0.391i)T \)
71 \( 1 + (0.278 - 0.960i)T \)
73 \( 1 + (0.428 + 0.903i)T \)
79 \( 1 + (0.948 - 0.316i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.996 - 0.0804i)T \)
97 \( 1 + (-0.845 + 0.534i)T \)
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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

−22.239184625167564245330470860042, −21.216122520561757793728648842357, −20.502624119332998047186269933585, −19.42784430641973727029936383649, −19.22239301728956865125889584772, −18.21111680482652877846420835228, −16.884605300141573793838433733645, −16.250351804848901206062946271, −16.02629585922640203369513895165, −15.26718843877441525252805118623, −14.15914253871679897512819165320, −13.63080668669110117969991876737, −12.97098975488102409264743206696, −11.54870659818312693337455152499, −10.65244392065273558159991370354, −9.645727180353533655242820930719, −9.092976136494352515662344125895, −8.193089063731772008135537539353, −7.66606451806037288917189590728, −6.66217883006049787703810378078, −5.52853842043749075823437789076, −4.66987482521160919363597563100, −3.72969901846396738354902071485, −3.28370596174757384386853406124, −1.240581631731380858837498263342, 0.205303754447111893070644791836, 1.74347180313084244377910291553, 2.536375780556407709832462489111, 3.46218981581138128938973291983, 3.92894267345716911617075847137, 5.47254260695445235944569084466, 6.62380666859541378141605881593, 7.57425716503078406184350034473, 8.32093056496130720628681873459, 8.96941282166077364871954798486, 10.04087721804468466867753118691, 10.702109859904665191880655405060, 11.85019308714368511634982582342, 12.49593664506272452738818555113, 12.89174678263052673836144053247, 13.89519390324697594927389997285, 14.913699325358883375362306832781, 15.37252167678968822497607828649, 16.569160253879438401213270375960, 17.83236703669426209371907745804, 18.3855143709904964879116958979, 18.94167253410259402503979399297, 19.55960695974285503958826676403, 20.3408002495750294006868618804, 20.82040855263872125608686684707

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