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Properties

Label 1-1480-1480.597-r0-0-0
Degree $1$
Conductor $1480$
Sign $-0.860 + 0.508i$
Analytic cond. $6.87309$
Root an. cond. $6.87309$
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  + (0.642 − 0.766i)3-s + (−0.342 − 0.939i)7-s + (−0.173 − 0.984i)9-s + (−0.5 − 0.866i)11-s + (−0.173 + 0.984i)13-s + (−0.173 − 0.984i)17-s + (−0.642 + 0.766i)19-s + (−0.939 − 0.342i)21-s + (−0.5 + 0.866i)23-s + (−0.866 − 0.5i)27-s + (−0.866 + 0.5i)29-s − i·31-s + (−0.984 − 0.173i)33-s + (0.642 + 0.766i)39-s + (−0.173 + 0.984i)41-s + ⋯
L(s)  = 1  + (0.642 − 0.766i)3-s + (−0.342 − 0.939i)7-s + (−0.173 − 0.984i)9-s + (−0.5 − 0.866i)11-s + (−0.173 + 0.984i)13-s + (−0.173 − 0.984i)17-s + (−0.642 + 0.766i)19-s + (−0.939 − 0.342i)21-s + (−0.5 + 0.866i)23-s + (−0.866 − 0.5i)27-s + (−0.866 + 0.5i)29-s − i·31-s + (−0.984 − 0.173i)33-s + (0.642 + 0.766i)39-s + (−0.173 + 0.984i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.860 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.860 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $-0.860 + 0.508i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (597, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ -0.860 + 0.508i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1530909251 - 0.5597095136i\)
\(L(\frac12)\) \(\approx\) \(-0.1530909251 - 0.5597095136i\)
\(L(1)\) \(\approx\) \(0.8330489912 - 0.4364763024i\)
\(L(1)\) \(\approx\) \(0.8330489912 - 0.4364763024i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
37 \( 1 \)
good3 \( 1 + (0.642 - 0.766i)T \)
7 \( 1 + (-0.342 - 0.939i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.173 + 0.984i)T \)
17 \( 1 + (-0.173 - 0.984i)T \)
19 \( 1 + (-0.642 + 0.766i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 - iT \)
41 \( 1 + (-0.173 + 0.984i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 + (0.342 - 0.939i)T \)
59 \( 1 + (0.342 - 0.939i)T \)
61 \( 1 + (0.984 + 0.173i)T \)
67 \( 1 + (-0.342 - 0.939i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + iT \)
79 \( 1 + (0.342 + 0.939i)T \)
83 \( 1 + (-0.984 + 0.173i)T \)
89 \( 1 + (0.342 - 0.939i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−21.06865126282020407167792298076, −20.37742805269795987073666632891, −19.66987582229010656564373629716, −19.03935440108917904107229009914, −18.081016418963247992270779639408, −17.38585796470830033043628782919, −16.405306000390490610016532095140, −15.600078227419285357393649056114, −15.08183790404639347554039386587, −14.68439204030922323353274252088, −13.4108286506544725700796171811, −12.81611710382586880248782794898, −12.09899779248468855584605063093, −10.85506110733703384352496948157, −10.29778558563599286413500384096, −9.5731991591060260443348385593, −8.66439943729371032285378478877, −8.19629769539532168541321924036, −7.15922770327089875212208802086, −6.043382589158202398269716281470, −5.20651202958735727945462572601, −4.454329369263057274839842855434, −3.45041918118662342055882371677, −2.56256339984857622039753745084, −1.95657884766284184923236028717, 0.18177639039042405990159864686, 1.426203687563114429612234589799, 2.32342302482088299486997808896, 3.40757247994936792977472801286, 3.96895755922577890970871857996, 5.24352529218957904074446426654, 6.34321294968471647131388521731, 6.93782599232208889819163625602, 7.77144112883241174762954850923, 8.40280759706334454474994128728, 9.44414862311034941411271360733, 9.9918947697724525476077697304, 11.25900358679167770041538401500, 11.71091500548050795582251892449, 12.97794452872549376102633339362, 13.312287250427884903603194016389, 14.10441183439870708443278483554, 14.63057537836068894037732133223, 15.765321351391804224794951084285, 16.55388791428185653439445804754, 17.13338508158115159962356242399, 18.30436115369944771808303265305, 18.67284642751927820613922439704, 19.517635200801570885551819327447, 20.076962369448552583520182153224

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