↑

Properties

Label 1-3503-3503.165-r0-0-0
Degree $1$
Conductor $3503$
Sign $-0.695 - 0.718i$
Analytic cond. $16.2678$
Root an. cond. $16.2678$
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  + (0.691 + 0.722i)2-s + (0.717 − 0.696i)3-s + (−0.0448 + 0.998i)4-s + (−0.884 + 0.467i)5-s + (0.999 + 0.0373i)6-s + (−0.599 + 0.800i)7-s + (−0.753 + 0.657i)8-s + (0.0299 − 0.999i)9-s + (−0.948 − 0.316i)10-s + (0.701 + 0.712i)11-s + (0.663 + 0.748i)12-s + (0.486 + 0.873i)13-s + (−0.992 + 0.119i)14-s + (−0.309 + 0.951i)15-s + (−0.995 − 0.0896i)16-s + (−0.427 + 0.904i)17-s + ⋯
L(s)  = 1  + (0.691 + 0.722i)2-s + (0.717 − 0.696i)3-s + (−0.0448 + 0.998i)4-s + (−0.884 + 0.467i)5-s + (0.999 + 0.0373i)6-s + (−0.599 + 0.800i)7-s + (−0.753 + 0.657i)8-s + (0.0299 − 0.999i)9-s + (−0.948 − 0.316i)10-s + (0.701 + 0.712i)11-s + (0.663 + 0.748i)12-s + (0.486 + 0.873i)13-s + (−0.992 + 0.119i)14-s + (−0.309 + 0.951i)15-s + (−0.995 − 0.0896i)16-s + (−0.427 + 0.904i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3503 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3503 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3503\)    =    \(31 \cdot 113\)
Sign: $-0.695 - 0.718i$
Analytic conductor: \(16.2678\)
Root analytic conductor: \(16.2678\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3503} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3503,\ (0:\ ),\ -0.695 - 0.718i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4046056852 + 0.9554518115i\)
\(L(\frac12)\) \(\approx\) \(-0.4046056852 + 0.9554518115i\)
\(L(1)\) \(\approx\) \(1.026718336 + 0.7266599685i\)
\(L(1)\) \(\approx\) \(1.026718336 + 0.7266599685i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 \)
113 \( 1 \)
good2 \( 1 + (0.691 + 0.722i)T \)
3 \( 1 + (0.717 - 0.696i)T \)
5 \( 1 + (-0.884 + 0.467i)T \)
7 \( 1 + (-0.599 + 0.800i)T \)
11 \( 1 + (0.701 + 0.712i)T \)
13 \( 1 + (0.486 + 0.873i)T \)
17 \( 1 + (-0.427 + 0.904i)T \)
19 \( 1 + (-0.985 - 0.171i)T \)
23 \( 1 + (-0.928 + 0.372i)T \)
29 \( 1 + (0.0224 - 0.999i)T \)
37 \( 1 + (0.399 + 0.916i)T \)
41 \( 1 + (0.635 + 0.772i)T \)
43 \( 1 + (0.480 + 0.877i)T \)
47 \( 1 + (-0.979 - 0.200i)T \)
53 \( 1 + (-0.251 + 0.967i)T \)
59 \( 1 + (-0.696 - 0.717i)T \)
61 \( 1 + (0.781 - 0.623i)T \)
67 \( 1 + (0.804 + 0.593i)T \)
71 \( 1 + (-0.933 - 0.358i)T \)
73 \( 1 + (-0.777 - 0.629i)T \)
79 \( 1 + (0.00747 + 0.999i)T \)
83 \( 1 + (0.971 - 0.237i)T \)
89 \( 1 + (-0.822 - 0.569i)T \)
97 \( 1 + (0.753 + 0.657i)T \)
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

−18.76718496057840537930510304259, −17.68049259372927971686154624558, −16.51444225212158851672317830680, −16.09166393895061103941836098248, −15.60778002492625020628389774022, −14.637529537577379572426675509378, −14.227247134626979585619029426656, −13.376999318312553938621504447009, −12.91242079893945656487327907471, −12.12000263878693898396592825598, −11.17015623440856993221201087903, −10.74413946145843301129626725837, −10.07537821812869163420787857825, −9.12975482113038540000958680407, −8.71173486669161240535038819034, −7.77768727582991722817794539428, −6.87668697240833859796551895058, −5.94057986115197305003166677711, −5.043970759072480378733602934763, −4.1942968276466927556381045370, −3.80672685096027862938881456119, −3.24399174216517715313888561369, −2.363375232924595005106106260858, −1.14709425108916559501241069359, −0.21257258863302899435713077189, 1.71422962433559692583297300908, 2.47122973818284800062355633467, 3.26435080801927862469611271980, 4.10035803777946249325557959584, 4.4100959131913995217067448766, 6.07174442929245966789935877816, 6.390694436089070259426689066783, 6.899772689353969127289702763579, 7.903399179950095035056570667002, 8.30035763291003335968197099716, 9.07789677002405720017978862118, 9.76708933052815500379042473777, 11.23862588826108215360809792383, 11.787386090589285956130487317922, 12.42287320042583278446255074719, 12.95901311152457939932992817192, 13.69989055656659520101434718384, 14.604905315294693837942256339166, 14.89147709648277173611475340442, 15.550608872507472195950722748176, 16.12539341050539953337669551225, 17.09964274721073193018712259135, 17.811823101039944711119516200, 18.54925470048133457716659995913, 19.22940214030749455296550421311

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