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Properties

Label 1-1480-1480.339-r0-0-0
Degree $1$
Conductor $1480$
Sign $0.763 + 0.646i$
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  + 3-s + 7-s + 9-s − 11-s + i·13-s − i·17-s + i·19-s + 21-s + i·23-s + 27-s + i·29-s + i·31-s − 33-s + i·39-s − 41-s + ⋯
L(s)  = 1  + 3-s + 7-s + 9-s − 11-s + i·13-s − i·17-s + i·19-s + 21-s + i·23-s + 27-s + i·29-s + i·31-s − 33-s + i·39-s − 41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.763 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.763 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $0.763 + 0.646i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ 0.763 + 0.646i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.333663044 + 0.8555893994i\)
\(L(\frac12)\) \(\approx\) \(2.333663044 + 0.8555893994i\)
\(L(1)\) \(\approx\) \(1.593407266 + 0.2079021968i\)
\(L(1)\) \(\approx\) \(1.593407266 + 0.2079021968i\)

Euler product

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

−20.42211493723364436677490367642, −20.107391553558756852329840501344, −19.02287417408152167583889816820, −18.436222811514135891852138740934, −17.68288488535390282795990076446, −16.93866050084045656013945048004, −15.68052322108289244976686195074, −15.2408485103341455797593431377, −14.66613997893685996242704786064, −13.71177336612547710658477515524, −13.101508229529833384475876401, −12.43260449510045981417995878930, −11.22459540914882338297776704226, −10.52851421862342952242519542355, −9.83350413201214061783281499942, −8.73154531399039528778225114155, −8.078218153641983790981815986666, −7.71576753125455036504178449838, −6.580327608799391458453687613763, −5.43040384027791310732453014006, −4.637285060121935491704547384357, −3.78707603883182139080629385670, −2.64180198375207300457042274074, −2.15059618626996589687956264412, −0.85201448304966234448695566495, 1.35304238361570800376003084149, 2.06355785378153748575756271306, 3.01544355358523852607771563840, 3.95571816382520734291000551480, 4.84979293380400631080649657621, 5.57707102112323423230025367102, 7.127860981055282616253285984031, 7.41386159180571330061809623029, 8.47218148265922489614520840790, 8.91757345200837309298316714991, 9.96922881861661648512607892910, 10.62329733722618753251672522446, 11.66694489841522223698720293940, 12.34058361561528854336542348855, 13.461827160687416192034830480920, 13.94736886868458286780963484829, 14.57241851210956777586632073740, 15.41399221980663540233190680431, 16.06999665779317103196997077133, 16.9222354882242232030609073176, 18.1452995044999944210668142922, 18.38715666741525420892580715846, 19.233683864476055010702127684243, 20.161660418917314696900383069273, 20.737090502295047194922401913345

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