Properties

Label 1-1480-1480.837-r0-0-0
Degree $1$
Conductor $1480$
Sign $0.999 + 0.00214i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00214i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00214i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $0.999 + 0.00214i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (837, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ 0.999 + 0.00214i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.180838048 + 0.002341481304i\)
\(L(\frac12)\) \(\approx\) \(2.180838048 + 0.002341481304i\)
\(L(1)\) \(\approx\) \(1.438157438 + 0.08818701537i\)
\(L(1)\) \(\approx\) \(1.438157438 + 0.08818701537i\)

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

−20.63670700109112651580853482543, −19.60944185530826253248586795210, −19.33895691959030373308446486427, −18.53012328838848570127241479738, −17.96987525051908055822798023286, −16.597222015820719433461082101343, −16.35926288034924784235484030492, −15.10158745952019838523181606669, −14.6770497293589297408425539338, −13.77339938018749745162219564208, −13.1892322175178336528109908183, −12.23009543134284835136659170483, −11.844921047551283781954968872160, −10.587691585121452951404778597398, −9.56946384122743414080433863353, −8.98710646221547915516448475405, −8.512825851262775771619540412016, −7.20109207543424085459383211210, −6.72577178780306145796275782900, −5.969942395237513764631195022710, −4.665182426043151310801458171614, −3.60453583553156184645121244555, −3.06439644514801434079132070915, −1.97286893668593013086530134584, −1.09121651645513640386111488815, 0.88231958820553609997794149344, 2.127110477719149986307115531448, 3.17716515245852893738351930149, 3.78593962293653689712811518571, 4.49232106403765161677905238110, 5.77601514830256447512409065237, 6.58086289297802209739145963654, 7.50633797048635109910756861078, 8.363659007849314132312415988925, 9.08983874846193771625767004630, 9.8414173057742965122954701892, 10.51339230622073801952361144507, 11.27947552090180556194287304158, 12.57210310275283774702277897788, 13.15491787019981643554350289991, 13.786453512973962438740473929835, 14.81586703852959953882937757971, 15.20317534669117687215425264614, 16.08016223458726116622424337799, 16.917607040267413775633675560944, 17.38864443486221994709839042031, 18.79489748156451253232054503891, 19.30303447294455181443600793889, 19.83670020865551540636007434278, 20.64013819945846171186257962413

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