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Properties

Label 1-1480-1480.363-r0-0-0
Degree $1$
Conductor $1480$
Sign $-0.377 - 0.925i$
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 + (0.984 + 0.173i)13-s + (−0.984 + 0.173i)17-s + (0.766 + 0.642i)19-s + (0.939 + 0.342i)21-s + (−0.866 − 0.5i)23-s + (0.866 + 0.5i)27-s + (0.5 + 0.866i)29-s + 31-s + (0.984 + 0.173i)33-s + (−0.766 + 0.642i)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 + (0.984 + 0.173i)13-s + (−0.984 + 0.173i)17-s + (0.766 + 0.642i)19-s + (0.939 + 0.342i)21-s + (−0.866 − 0.5i)23-s + (0.866 + 0.5i)27-s + (0.5 + 0.866i)29-s + 31-s + (0.984 + 0.173i)33-s + (−0.766 + 0.642i)39-s + (0.173 − 0.984i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.377 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.377 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $-0.377 - 0.925i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (363, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ -0.377 - 0.925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3169189601 - 0.4715532174i\)
\(L(\frac12)\) \(\approx\) \(0.3169189601 - 0.4715532174i\)
\(L(1)\) \(\approx\) \(0.7247128361 + 0.02787466361i\)
\(L(1)\) \(\approx\) \(0.7247128361 + 0.02787466361i\)

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 \( 1 + (0.984 + 0.173i)T \)
17 \( 1 + (-0.984 + 0.173i)T \)
19 \( 1 + (0.766 + 0.642i)T \)
23 \( 1 + (-0.866 - 0.5i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + T \)
41 \( 1 + (0.173 - 0.984i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 + (-0.342 + 0.939i)T \)
59 \( 1 + (-0.939 - 0.342i)T \)
61 \( 1 + (0.173 - 0.984i)T \)
67 \( 1 + (0.342 + 0.939i)T \)
71 \( 1 + (-0.766 - 0.642i)T \)
73 \( 1 + iT \)
79 \( 1 + (0.939 - 0.342i)T \)
83 \( 1 + (-0.984 + 0.173i)T \)
89 \( 1 + (-0.939 - 0.342i)T \)
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

−20.97582245119439820154287602914, −19.917620397183488232206914494392, −19.396565145212589407548404577039, −18.376922481682139172424993058892, −17.956604809329234286159077793486, −17.5132905500332097138837268889, −16.17299608435225520052455738989, −15.80364394367339043490727037043, −15.035348914759321535650056989943, −13.73775454384871150739899677046, −13.25996260681353132801258005481, −12.52726300978692648260898678255, −11.70075814352569775577510670670, −11.240465994775360960354463917469, −10.13578890539615111546101749854, −9.36467216250117232927879772020, −8.30337636605429704383154513010, −7.70012952715506654734983583071, −6.569822240385145185944730191591, −6.159545287209185136077389202449, −5.19019261046362187325423806791, −4.44663231764328216761091850722, −2.9651075511286822051504699708, −2.249675143726030009526204997046, −1.234398153212955142990276046869, 0.25483072165418139362304447413, 1.37287292877496516991282462435, 2.98137166357359843791843168053, 3.760613603371001345070944126149, 4.41563342375582267985287903628, 5.4609235573311725412322641956, 6.23126201348223210869056620134, 6.89175632842253668979668889657, 8.128737941132194380832345078205, 8.837647779878284171734610263403, 9.85164263930692296813322162244, 10.559526747663436985479900650019, 10.99917305740359313347471041115, 11.87832813536543878693090991893, 12.79922875933233420758090142279, 13.77882519592242839980328833450, 14.16776619810120627723928314787, 15.536641273840021121353340453861, 15.952729405907269116523426808079, 16.5372221190190824026434081320, 17.318256842747119865526100673713, 18.106462854643637273569639743910, 18.79633805302039190339753029217, 19.91560709606125030679471654355, 20.50596672909024137344827637514

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