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Properties

Label 1-683-683.60-r1-0-0
Degree $1$
Conductor $683$
Sign $-0.970 - 0.241i$
Analytic cond. $73.3985$
Root an. cond. $73.3985$
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  + (−0.841 + 0.540i)2-s + (0.528 + 0.848i)3-s + (0.415 − 0.909i)4-s + (0.721 + 0.692i)5-s + (−0.903 − 0.427i)6-s + (−0.00460 + 0.999i)7-s + (0.142 + 0.989i)8-s + (−0.440 + 0.897i)9-s + (−0.981 − 0.192i)10-s + (0.964 − 0.264i)11-s + (0.991 − 0.128i)12-s + (0.977 − 0.210i)13-s + (−0.536 − 0.843i)14-s + (−0.205 + 0.978i)15-s + (−0.654 − 0.755i)16-s + (−0.381 + 0.924i)17-s + ⋯
L(s)  = 1  + (−0.841 + 0.540i)2-s + (0.528 + 0.848i)3-s + (0.415 − 0.909i)4-s + (0.721 + 0.692i)5-s + (−0.903 − 0.427i)6-s + (−0.00460 + 0.999i)7-s + (0.142 + 0.989i)8-s + (−0.440 + 0.897i)9-s + (−0.981 − 0.192i)10-s + (0.964 − 0.264i)11-s + (0.991 − 0.128i)12-s + (0.977 − 0.210i)13-s + (−0.536 − 0.843i)14-s + (−0.205 + 0.978i)15-s + (−0.654 − 0.755i)16-s + (−0.381 + 0.924i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 683 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 683 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(683\)
Sign: $-0.970 - 0.241i$
Analytic conductor: \(73.3985\)
Root analytic conductor: \(73.3985\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{683} (60, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 683,\ (1:\ ),\ -0.970 - 0.241i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2246970570 + 1.834335081i\)
\(L(\frac12)\) \(\approx\) \(-0.2246970570 + 1.834335081i\)
\(L(1)\) \(\approx\) \(0.7007582753 + 0.7798750418i\)
\(L(1)\) \(\approx\) \(0.7007582753 + 0.7798750418i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad683 \( 1 \)
good2 \( 1 + (-0.841 + 0.540i)T \)
3 \( 1 + (0.528 + 0.848i)T \)
5 \( 1 + (0.721 + 0.692i)T \)
7 \( 1 + (-0.00460 + 0.999i)T \)
11 \( 1 + (0.964 - 0.264i)T \)
13 \( 1 + (0.977 - 0.210i)T \)
17 \( 1 + (-0.381 + 0.924i)T \)
19 \( 1 + (0.633 - 0.773i)T \)
23 \( 1 + (-0.932 - 0.360i)T \)
29 \( 1 + (-0.883 - 0.469i)T \)
31 \( 1 + (0.668 + 0.743i)T \)
37 \( 1 + (0.874 + 0.485i)T \)
41 \( 1 + (-0.575 + 0.818i)T \)
43 \( 1 + (-0.995 + 0.0919i)T \)
47 \( 1 + (0.865 + 0.501i)T \)
53 \( 1 + (-0.793 - 0.608i)T \)
59 \( 1 + (-0.0874 + 0.996i)T \)
61 \( 1 + (-0.990 - 0.137i)T \)
67 \( 1 + (0.918 - 0.394i)T \)
71 \( 1 + (0.956 - 0.290i)T \)
73 \( 1 + (-0.945 - 0.325i)T \)
79 \( 1 + (-0.0230 - 0.999i)T \)
83 \( 1 + (0.675 + 0.737i)T \)
89 \( 1 + (-0.590 + 0.807i)T \)
97 \( 1 + (0.911 + 0.411i)T \)
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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

−21.880357997847358155630517373928, −20.68275030487049904701034002132, −20.34267955753901657469901989788, −19.87293440427091010520178813975, −18.67289295366644121855531312054, −18.140630612925219301315262901684, −17.247149484155770809438308033800, −16.73379951915472174632642234165, −15.75379249245866850891853343462, −14.15460782127368506323711170864, −13.68397409055048132002666554534, −12.9017282507938216808826982449, −11.98201944250048230378540552216, −11.2567876696346695068593451760, −9.93715081388657261446405649586, −9.37724497626002645475134356448, −8.55901664803780897434556488011, −7.67374874711242837378731253589, −6.84932314046832631511840686827, −5.967642872299015386250935306956, −4.203189624259830913124336922, −3.41114963741883908166706878681, −1.99867848771203558066389828351, −1.37068865498946868701949145662, −0.52663249283289521890315758349, 1.50027800376677296120269959067, 2.45803390188870073158594036065, 3.482379760807495567076428992787, 4.91340773494934203265178635881, 6.04333669086839907443387555062, 6.39188865663552432254250286705, 7.89860318496970267090181714114, 8.71805602326832816343386586657, 9.3174869226573219534674322004, 10.0700696579405677211875320556, 10.96365912777062687673866289196, 11.63857183537787351957524801824, 13.38325357483523484212714365015, 14.1458879922159727979370781453, 14.97522741929769961671504439836, 15.439231364573417274637654928352, 16.33774139424067190109615424652, 17.21480260805492208908693404018, 18.06463068518583764813295211361, 18.75469148422621215526164833348, 19.60445265202982438988834408714, 20.36508405633025162290966905447, 21.42206140094367460433376787128, 22.028628659805095692508301113133, 22.73834462830568587171981489581

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