Properties

Label 1-1480-1480.717-r0-0-0
Degree $1$
Conductor $1480$
Sign $0.166 + 0.986i$
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  + (0.866 + 0.5i)3-s + (−0.866 − 0.5i)7-s + (0.5 + 0.866i)9-s + 11-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)17-s + (0.866 + 0.5i)19-s + (−0.5 − 0.866i)21-s − 23-s i·27-s i·29-s + i·31-s + (0.866 + 0.5i)33-s + (−0.866 + 0.5i)39-s + (0.5 − 0.866i)41-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)3-s + (−0.866 − 0.5i)7-s + (0.5 + 0.866i)9-s + 11-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)17-s + (0.866 + 0.5i)19-s + (−0.5 − 0.866i)21-s − 23-s i·27-s i·29-s + i·31-s + (0.866 + 0.5i)33-s + (−0.866 + 0.5i)39-s + (0.5 − 0.866i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.166 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.166 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $0.166 + 0.986i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (717, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ 0.166 + 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.406762846 + 1.189114994i\)
\(L(\frac12)\) \(\approx\) \(1.406762846 + 1.189114994i\)
\(L(1)\) \(\approx\) \(1.260891672 + 0.3628786416i\)
\(L(1)\) \(\approx\) \(1.260891672 + 0.3628786416i\)

Euler product

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

−20.1386394315624906690839904128, −19.7559418779449345436213887447, −19.2134050789337560634121538006, −18.31372279055974399384714043286, −17.64043531511866834550049495049, −16.79594600524880330504331897868, −15.70102766005285183110731659941, −15.21625075269431091842243757481, −14.46389736217143946612579603601, −13.56514182376281680383805358151, −12.982858833181935175019592277859, −12.216546025903658600688965627476, −11.57972158618360810679296110178, −10.21561231240292165064036000374, −9.556049058799977575724704877902, −8.94217141712086813333396810622, −8.03963119726971288187247337341, −7.314765679166804870299537638051, −6.33832287336505487085741568017, −5.82692666917849179700443006591, −4.33912909928595572411483748158, −3.565310375427483612065360515574, −2.70596315070416651122623229258, −1.93298263780517089508155752435, −0.64225386867857437976582170262, 1.25217898632571169527895657460, 2.35391116962263399427216017638, 3.29209468033807067991704108034, 4.00087747280424282806949553817, 4.71705957032332730321736761991, 5.9293164251096330234737293020, 7.046839521392112168430725863071, 7.38297648676158612520770282097, 8.71802938409879537599880018455, 9.2733076124052402162100301874, 9.861328726357868434327799523166, 10.6579533865955318629565006586, 11.7266167525097291841705659234, 12.46732728725565452120275066789, 13.48250063312675443848828798407, 14.25187203747154931033346909178, 14.400108222333423205261146752882, 15.843899124285726135798003443170, 16.09599137500322593727434114605, 16.84561922332814067870130665264, 17.858897037358311135011503725096, 18.823274840695297343393801253176, 19.566569520488148925860312621990, 19.97079830826323535005597609683, 20.66694918814728122677102986203

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