Properties

Label 1-1480-1480.469-r0-0-0
Degree $1$
Conductor $1480$
Sign $-0.850 - 0.526i$
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  + (0.173 + 0.984i)3-s + (−0.766 + 0.642i)7-s + (−0.939 + 0.342i)9-s + (0.5 − 0.866i)11-s + (0.939 + 0.342i)13-s + (−0.939 + 0.342i)17-s + (0.173 + 0.984i)19-s + (−0.766 − 0.642i)21-s + (−0.5 − 0.866i)23-s + (−0.5 − 0.866i)27-s + (−0.5 + 0.866i)29-s − 31-s + (0.939 + 0.342i)33-s + (−0.173 + 0.984i)39-s + (−0.939 − 0.342i)41-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)3-s + (−0.766 + 0.642i)7-s + (−0.939 + 0.342i)9-s + (0.5 − 0.866i)11-s + (0.939 + 0.342i)13-s + (−0.939 + 0.342i)17-s + (0.173 + 0.984i)19-s + (−0.766 − 0.642i)21-s + (−0.5 − 0.866i)23-s + (−0.5 − 0.866i)27-s + (−0.5 + 0.866i)29-s − 31-s + (0.939 + 0.342i)33-s + (−0.173 + 0.984i)39-s + (−0.939 − 0.342i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $-0.850 - 0.526i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ -0.850 - 0.526i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1286722719 + 0.4526525016i\)
\(L(\frac12)\) \(\approx\) \(-0.1286722719 + 0.4526525016i\)
\(L(1)\) \(\approx\) \(0.7420634002 + 0.3816075989i\)
\(L(1)\) \(\approx\) \(0.7420634002 + 0.3816075989i\)

Euler product

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

−20.01342558531518521426186009767, −19.68861581322082812239649333924, −18.6803351773043725618899964018, −17.96225552522068032084254530599, −17.40352789415632954387859215776, −16.594099679725295524213308274980, −15.60671649190513101468360346951, −14.963370854345649160418538136474, −13.8243729464518974925343445245, −13.34003480685239062121995971830, −12.87822935200770930869686435574, −11.77438808384844966529172894846, −11.25981231536428853485989775797, −10.139097221947910361803334385899, −9.30235294121379435096035189964, −8.58723421301336064826866405348, −7.51359265465877146328332569173, −6.94303853348869825251237099153, −6.331209858941144288556998978789, −5.3360007261065673098353210239, −4.07812334823436002742118590788, −3.341575001130705225807344414017, −2.26080099050263155283129298972, −1.372363466932432599068270762217, −0.16705101586113899152329976812, 1.64550308615103766849268349989, 2.804916767131143430401445359830, 3.64105510401770164679156538531, 4.16922892306844127367545274394, 5.49023080686934040526657287619, 6.00765360859611967139007228909, 6.84971991897531620062660572320, 8.383432036133659612992524209896, 8.727712992744117956923793860782, 9.45101530219047879645225905545, 10.374080480455598678709203561628, 11.046362669127083066566851008010, 11.813208679221733059157822047, 12.75692340800126693232016353164, 13.66440034255835533729793720612, 14.36664243920225089963451415840, 15.1581609301499556994489082780, 15.95554748604123779829020670970, 16.41453301694414615493124497161, 17.034034426420925649406880978628, 18.369920379710688928449863788223, 18.74971575190441790940411255349, 19.8226217858274041714330963111, 20.2787928049282944924858458511, 21.230495392846064601626936743111

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