↑

Properties

Label 1-1007-1007.7-r0-0-0
Degree $1$
Conductor $1007$
Sign $-0.120 - 0.992i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.799 − 0.600i)2-s + (0.845 − 0.534i)3-s + (0.278 + 0.960i)4-s + (0.996 − 0.0804i)5-s + (−0.996 − 0.0804i)6-s + (0.120 − 0.992i)7-s + (0.354 − 0.935i)8-s + (0.428 − 0.903i)9-s + (−0.845 − 0.534i)10-s + (−0.748 − 0.663i)11-s + (0.748 + 0.663i)12-s + (0.692 + 0.721i)13-s + (−0.692 + 0.721i)14-s + (0.799 − 0.600i)15-s + (−0.845 + 0.534i)16-s + (0.987 + 0.160i)17-s + ⋯
L(s)  = 1  + (−0.799 − 0.600i)2-s + (0.845 − 0.534i)3-s + (0.278 + 0.960i)4-s + (0.996 − 0.0804i)5-s + (−0.996 − 0.0804i)6-s + (0.120 − 0.992i)7-s + (0.354 − 0.935i)8-s + (0.428 − 0.903i)9-s + (−0.845 − 0.534i)10-s + (−0.748 − 0.663i)11-s + (0.748 + 0.663i)12-s + (0.692 + 0.721i)13-s + (−0.692 + 0.721i)14-s + (0.799 − 0.600i)15-s + (−0.845 + 0.534i)16-s + (0.987 + 0.160i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1007 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.120 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1007 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.120 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1007\)    =    \(19 \cdot 53\)
Sign: $-0.120 - 0.992i$
Analytic conductor: \(4.67648\)
Root analytic conductor: \(4.67648\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1007} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1007,\ (0:\ ),\ -0.120 - 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.210957717 - 1.366803182i\)
\(L(\frac12)\) \(\approx\) \(1.210957717 - 1.366803182i\)
\(L(1)\) \(\approx\) \(1.064122560 - 0.6445161838i\)
\(L(1)\) \(\approx\) \(1.064122560 - 0.6445161838i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
53 \( 1 \)
good2 \( 1 + (-0.799 - 0.600i)T \)
3 \( 1 + (0.845 - 0.534i)T \)
5 \( 1 + (0.996 - 0.0804i)T \)
7 \( 1 + (0.120 - 0.992i)T \)
11 \( 1 + (-0.748 - 0.663i)T \)
13 \( 1 + (0.692 + 0.721i)T \)
17 \( 1 + (0.987 + 0.160i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.948 + 0.316i)T \)
31 \( 1 + (0.748 - 0.663i)T \)
37 \( 1 + (0.885 + 0.464i)T \)
41 \( 1 + (-0.948 + 0.316i)T \)
43 \( 1 + (-0.0402 + 0.999i)T \)
47 \( 1 + (-0.996 - 0.0804i)T \)
59 \( 1 + (0.428 + 0.903i)T \)
61 \( 1 + (-0.987 + 0.160i)T \)
67 \( 1 + (-0.278 - 0.960i)T \)
71 \( 1 + (0.0402 - 0.999i)T \)
73 \( 1 + (-0.987 - 0.160i)T \)
79 \( 1 + (0.919 + 0.391i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.632 + 0.774i)T \)
97 \( 1 + (-0.996 + 0.0804i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.62082332856016196092604456050, −20.83280644574700084377991899656, −20.48488461047593628771563659885, −19.31233585082069140888546698305, −18.53739959864877053448058203299, −18.08909856794493824434760393381, −17.204000000072302393717462040914, −16.19274071170177254062611300993, −15.58254979529480767726039619671, −14.87656630719653053937843101610, −14.2523567737151265031938036529, −13.364839590228291021938004272437, −12.43473158415891802919071155305, −11.01748222815773207942611666425, −10.06869799802636995898450614229, −9.9003870568185556661184766741, −8.701971228315092141916131210885, −8.39672033854033351308993972851, −7.36079855496122480585748323871, −6.26379377742984884395369713844, −5.38024149407150077205451961850, −4.80410819673568075322958583922, −2.98714323243225200258210579157, −2.38523629413631784069329898005, −1.36115950800829198048246463212, 1.08165108768285346130298568498, 1.53373240003173641365174990767, 2.793074447461401439485325459751, 3.38432188021388462422903463879, 4.5652429534744877081148811386, 6.11450765677269359303769494398, 6.89544291844081684047181628952, 7.88279886497839159826495317053, 8.405943321447016667847597773712, 9.43089863937968824842800535689, 9.98841581078005230936429576071, 10.82159584250139730402183274576, 11.75793416888140751268937277651, 12.86430234911745568527030511291, 13.564115182984923913039464334049, 13.82287697046064944016621685366, 15.03804962925206970467349536369, 16.31469312734650153451384306494, 16.8127632936971864291380083985, 17.82316383182325509089611091297, 18.327225460242475544924380350791, 19.11791771329857686450246373117, 19.76582056260804729934875333348, 20.76748763471729847329997100755, 21.08390114337861059278756393324

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