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Properties

Label 1-739-739.293-r0-0-0
Degree $1$
Conductor $739$
Sign $-0.495 - 0.868i$
Analytic cond. $3.43190$
Root an. cond. $3.43190$
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  + (0.817 − 0.575i)2-s + (−0.859 + 0.511i)3-s + (0.338 − 0.941i)4-s + (−0.859 − 0.511i)5-s + (−0.409 + 0.912i)6-s + (0.817 − 0.575i)7-s + (−0.264 − 0.964i)8-s + (0.477 − 0.878i)9-s + (−0.997 + 0.0765i)10-s + (0.896 + 0.443i)11-s + (0.190 + 0.981i)12-s + (−0.973 + 0.227i)13-s + (0.338 − 0.941i)14-s + 15-s + (−0.771 − 0.636i)16-s + (0.953 − 0.301i)17-s + ⋯
L(s)  = 1  + (0.817 − 0.575i)2-s + (−0.859 + 0.511i)3-s + (0.338 − 0.941i)4-s + (−0.859 − 0.511i)5-s + (−0.409 + 0.912i)6-s + (0.817 − 0.575i)7-s + (−0.264 − 0.964i)8-s + (0.477 − 0.878i)9-s + (−0.997 + 0.0765i)10-s + (0.896 + 0.443i)11-s + (0.190 + 0.981i)12-s + (−0.973 + 0.227i)13-s + (0.338 − 0.941i)14-s + 15-s + (−0.771 − 0.636i)16-s + (0.953 − 0.301i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 739 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.495 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 739 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.495 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(739\)
Sign: $-0.495 - 0.868i$
Analytic conductor: \(3.43190\)
Root analytic conductor: \(3.43190\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{739} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 739,\ (0:\ ),\ -0.495 - 0.868i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7750703352 - 1.333650054i\)
\(L(\frac12)\) \(\approx\) \(0.7750703352 - 1.333650054i\)
\(L(1)\) \(\approx\) \(1.065884802 - 0.6151163267i\)
\(L(1)\) \(\approx\) \(1.065884802 - 0.6151163267i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad739 \( 1 \)
good2 \( 1 + (0.817 - 0.575i)T \)
3 \( 1 + (-0.859 + 0.511i)T \)
5 \( 1 + (-0.859 - 0.511i)T \)
7 \( 1 + (0.817 - 0.575i)T \)
11 \( 1 + (0.896 + 0.443i)T \)
13 \( 1 + (-0.973 + 0.227i)T \)
17 \( 1 + (0.953 - 0.301i)T \)
19 \( 1 + (0.988 + 0.152i)T \)
23 \( 1 + (0.817 - 0.575i)T \)
29 \( 1 + (-0.665 - 0.746i)T \)
31 \( 1 + (-0.973 + 0.227i)T \)
37 \( 1 + (-0.771 - 0.636i)T \)
41 \( 1 + (-0.114 + 0.993i)T \)
43 \( 1 + (0.0383 - 0.999i)T \)
47 \( 1 + (0.606 - 0.795i)T \)
53 \( 1 + (-0.771 - 0.636i)T \)
59 \( 1 + (0.0383 + 0.999i)T \)
61 \( 1 + (-0.409 - 0.912i)T \)
67 \( 1 + (0.477 - 0.878i)T \)
71 \( 1 + (0.606 + 0.795i)T \)
73 \( 1 + (-0.543 + 0.839i)T \)
79 \( 1 + (0.720 - 0.693i)T \)
83 \( 1 + (0.720 - 0.693i)T \)
89 \( 1 + (-0.973 - 0.227i)T \)
97 \( 1 + (0.817 - 0.575i)T \)
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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

−22.65078539771475942234941129213, −22.2297455370325224038347451667, −21.6152745612072904991535148722, −20.468181103842731315182981769, −19.38636742144781707897257357736, −18.64240009500855997887839849517, −17.695205498793788801084747817247, −16.99997682584636593051802927711, −16.23724660957871021785543441898, −15.29683045312420238405222774697, −14.59640515601895499454439340235, −13.9224115668666347116534503979, −12.60181012826693759243701399488, −12.08255913328217812562555825830, −11.44542584828438498143678621312, −10.80685365613841972116492715413, −9.15075935998215843837542426280, −7.864798323832128349649295551245, −7.47255458761462039647948401461, −6.60113323442090474861704368265, −5.50824224293842944684695835689, −5.01323281535732670380839158035, −3.823505316535873586278856801405, −2.83284504422605173621045245675, −1.43134221249864633615457997107, 0.68279823446724187549720039870, 1.66618234073030561087176292513, 3.42784339632379067390851549429, 4.125694439668626900263384736477, 4.92009558213782994595254615870, 5.42926543365320114732059202273, 6.90460940947874725328481694289, 7.497022099593996374534591967487, 9.127778177159711045499826166123, 9.897971750963015674290410520836, 10.85951629726349028189235853807, 11.664334716422695397162339513191, 12.03257579040232645885943559073, 12.82167626345636382353597888052, 14.194313994150213075019428566822, 14.76113144856036223363593592592, 15.54632845759510682301211517931, 16.62579364026355695764898428092, 17.01899757911244111284095125751, 18.252946249801781762046050159420, 19.19458515605423319181183037338, 20.19644432622635466505835393791, 20.58537776141812795098498781381, 21.44400208628687639067096252545, 22.33875235089438560945308337788

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