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Properties

Label 1-1480-1480.1269-r0-0-0
Degree $1$
Conductor $1480$
Sign $-0.729 + 0.683i$
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  + (−0.5 − 0.866i)3-s + (0.5 + 0.866i)7-s + (−0.5 + 0.866i)9-s − 11-s + (0.5 + 0.866i)13-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.5 − 0.866i)21-s + 23-s + 27-s + 29-s − 31-s + (0.5 + 0.866i)33-s + (0.5 − 0.866i)39-s + (−0.5 − 0.866i)41-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)3-s + (0.5 + 0.866i)7-s + (−0.5 + 0.866i)9-s − 11-s + (0.5 + 0.866i)13-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.5 − 0.866i)21-s + 23-s + 27-s + 29-s − 31-s + (0.5 + 0.866i)33-s + (0.5 − 0.866i)39-s + (−0.5 − 0.866i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.729 + 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.729 + 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $-0.729 + 0.683i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (1269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ -0.729 + 0.683i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1282030053 + 0.3242722177i\)
\(L(\frac12)\) \(\approx\) \(0.1282030053 + 0.3242722177i\)
\(L(1)\) \(\approx\) \(0.7346486735 + 0.01757695584i\)
\(L(1)\) \(\approx\) \(0.7346486735 + 0.01757695584i\)

Euler product

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

−20.51710894809338461083743026501, −19.974135866895479212362864982145, −18.69995089424796506210748108940, −17.91262775120202183342675685748, −17.44593515345570167705594137716, −16.42932500460763241477496352034, −16.05992573748339494779118751843, −15.079337780184147661027867974, −14.55602408619804606217495660872, −13.4545851321709184587006501667, −12.88131834358562110032211171574, −11.72025645590125055399679017959, −11.01987859242272327174228110674, −10.429844512424568753893544681708, −9.87540804334821337716155960084, −8.716215284709488448408960460677, −8.02098048489408556989874754080, −7.049668782276868368286666037106, −6.10415538029871116126400087637, −5.0986906203542467341830060627, −4.69550024755669309105934792375, −3.595685689210069721839675340772, −2.86948217001321470292461675038, −1.366981714856338164184892240936, −0.140316433629272052398773679503, 1.46582054668694221369086622194, 2.15506270052824285493091463263, 3.058097928089451833554544487765, 4.58082009679044742344268058218, 5.18349622626965211043344731594, 6.12996384566214724502757023328, 6.75798148149448425182286628813, 7.71683601190036219151086388590, 8.54681323775306989852942664920, 9.05977204080199848658596845710, 10.5561325264288962306001876472, 11.048057880070086614973327325543, 11.79888506851540914580893460081, 12.628824102681577987653153571002, 13.20920364895054567527245282581, 13.96844612201063210054842290649, 15.000541157898644713544735180608, 15.619070405707505658682073888353, 16.56969231253844441639253084111, 17.342931188654757398126691363787, 18.05783923113335503619570924795, 18.60651223757661291078833194514, 19.23582183863483917254106365902, 20.0404942442650434598316671659, 21.25391530376411015779674509332

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