Properties

Label 1-740-740.679-r0-0-0
Degree $1$
Conductor $740$
Sign $-0.142 - 0.989i$
Analytic cond. $3.43654$
Root an. cond. $3.43654$
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  + (0.939 − 0.342i)3-s + (0.173 − 0.984i)7-s + (0.766 − 0.642i)9-s + (−0.5 − 0.866i)11-s + (0.642 − 0.766i)13-s + (−0.642 − 0.766i)17-s + (0.342 + 0.939i)19-s + (−0.173 − 0.984i)21-s + (−0.866 − 0.5i)23-s + (0.5 − 0.866i)27-s + (−0.866 + 0.5i)29-s i·31-s + (−0.766 − 0.642i)33-s + (0.342 − 0.939i)39-s + (−0.766 − 0.642i)41-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)3-s + (0.173 − 0.984i)7-s + (0.766 − 0.642i)9-s + (−0.5 − 0.866i)11-s + (0.642 − 0.766i)13-s + (−0.642 − 0.766i)17-s + (0.342 + 0.939i)19-s + (−0.173 − 0.984i)21-s + (−0.866 − 0.5i)23-s + (0.5 − 0.866i)27-s + (−0.866 + 0.5i)29-s i·31-s + (−0.766 − 0.642i)33-s + (0.342 − 0.939i)39-s + (−0.766 − 0.642i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.142 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.142 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(740\)    =    \(2^{2} \cdot 5 \cdot 37\)
Sign: $-0.142 - 0.989i$
Analytic conductor: \(3.43654\)
Root analytic conductor: \(3.43654\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{740} (679, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 740,\ (0:\ ),\ -0.142 - 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.226721946 - 1.415318406i\)
\(L(\frac12)\) \(\approx\) \(1.226721946 - 1.415318406i\)
\(L(1)\) \(\approx\) \(1.295732080 - 0.5480197438i\)
\(L(1)\) \(\approx\) \(1.295732080 - 0.5480197438i\)

Euler product

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

−22.42657040434539861349990118811, −21.78094471056290452099093512291, −21.09480243156564873250580160833, −20.31097337392529818281654229887, −19.57997405994463381457390220687, −18.64299510900983145378355006547, −18.109677163710350677721401588945, −16.98817550775437901707502263452, −15.79808438944673472999169242708, −15.369495047854188011782536120293, −14.72346118972107043877298461949, −13.58987926646622363589162625866, −13.07003027174598449569387335223, −11.94127929893705490410173600273, −11.08203050366949614247980950739, −9.95549979157342378866180277017, −9.25161786270873495520718271137, −8.512519781319723536230567682474, −7.69152879895176767919742937762, −6.639501082049634187707587927777, −5.46923340699920664379450196894, −4.47694042887429948901433038038, −3.62804428555366478338859125895, −2.32141344922118611835936262835, −1.87798935153686225877315368212, 0.76818748419330512053910742011, 1.90392473930350725396747701789, 3.190934189903519626395269615938, 3.7348737597209947348338260003, 4.97813955208610249943420865998, 6.19843295178505454363411227959, 7.1644242031768449929647719221, 8.02462563445040354844127038844, 8.54751727862655577233672140333, 9.73608724650996822650733279626, 10.517570450313985583077685388895, 11.38963607246261306667908381566, 12.65442221810674597162556587728, 13.33824373882144845369445597, 14.01865663089893258897954367190, 14.63493426122035893736241474005, 15.90224370572982272824595569754, 16.271868980546511250769543810233, 17.65631266453670405794441324913, 18.273551465925256253584435139829, 19.044302161804177665839269960163, 20.065012860518335874795404465876, 20.48691285931794485209334941417, 21.15691598968600327974761470418, 22.31258380773524610462760324875

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