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Properties

Label 1-1340-1340.339-r1-0-0
Degree $1$
Conductor $1340$
Sign $-0.498 - 0.867i$
Analytic cond. $144.002$
Root an. cond. $144.002$
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  + (0.415 + 0.909i)3-s + (−0.327 − 0.945i)7-s + (−0.654 + 0.755i)9-s + (−0.235 + 0.971i)11-s + (0.888 + 0.458i)13-s + (−0.928 + 0.371i)17-s + (0.327 − 0.945i)19-s + (0.723 − 0.690i)21-s + (0.580 − 0.814i)23-s + (−0.959 − 0.281i)27-s + (−0.5 + 0.866i)29-s + (0.888 − 0.458i)31-s + (−0.981 + 0.189i)33-s + (0.5 + 0.866i)37-s + (−0.0475 + 0.998i)39-s + ⋯
L(s)  = 1  + (0.415 + 0.909i)3-s + (−0.327 − 0.945i)7-s + (−0.654 + 0.755i)9-s + (−0.235 + 0.971i)11-s + (0.888 + 0.458i)13-s + (−0.928 + 0.371i)17-s + (0.327 − 0.945i)19-s + (0.723 − 0.690i)21-s + (0.580 − 0.814i)23-s + (−0.959 − 0.281i)27-s + (−0.5 + 0.866i)29-s + (0.888 − 0.458i)31-s + (−0.981 + 0.189i)33-s + (0.5 + 0.866i)37-s + (−0.0475 + 0.998i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.498 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.498 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1340\)    =    \(2^{2} \cdot 5 \cdot 67\)
Sign: $-0.498 - 0.867i$
Analytic conductor: \(144.002\)
Root analytic conductor: \(144.002\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1340} (339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1340,\ (1:\ ),\ -0.498 - 0.867i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03985166543 - 0.06884587818i\)
\(L(\frac12)\) \(\approx\) \(0.03985166543 - 0.06884587818i\)
\(L(1)\) \(\approx\) \(0.9456025352 + 0.2798920657i\)
\(L(1)\) \(\approx\) \(0.9456025352 + 0.2798920657i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
67 \( 1 \)
good3 \( 1 + (0.415 + 0.909i)T \)
7 \( 1 + (-0.327 - 0.945i)T \)
11 \( 1 + (-0.235 + 0.971i)T \)
13 \( 1 + (0.888 + 0.458i)T \)
17 \( 1 + (-0.928 + 0.371i)T \)
19 \( 1 + (0.327 - 0.945i)T \)
23 \( 1 + (0.580 - 0.814i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (0.888 - 0.458i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.786 - 0.618i)T \)
43 \( 1 + (-0.142 + 0.989i)T \)
47 \( 1 + (-0.995 - 0.0950i)T \)
53 \( 1 + (0.142 + 0.989i)T \)
59 \( 1 + (-0.841 - 0.540i)T \)
61 \( 1 + (0.235 + 0.971i)T \)
71 \( 1 + (-0.928 - 0.371i)T \)
73 \( 1 + (-0.235 - 0.971i)T \)
79 \( 1 + (-0.0475 - 0.998i)T \)
83 \( 1 + (0.723 + 0.690i)T \)
89 \( 1 + (0.415 - 0.909i)T \)
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

−20.942478585036103226248493837837, −20.117429635153568867984410670711, −19.26835615504299547715444122421, −18.72086311601335876109145981051, −18.15847686780823946022373786621, −17.40270642137669237616538005062, −16.28271201828736171525455657291, −15.59497569697998419490261396215, −14.878900465417324165193399991512, −13.82308459722250601714871649316, −13.33445985319944594267622370279, −12.65519027752084760852737330810, −11.66503407399962928889796346023, −11.21944400862998318387900172036, −9.929026178386324350107953493733, −8.98276640059553061743519092306, −8.43221708987170652401942857852, −7.72554379943038601570318002550, −6.59285125233763488025992582276, −5.98243714115334703731989980665, −5.26665666625390029640355545356, −3.66628301340634790423370674322, −3.035368656512767553520822805956, −2.12488454336345746482137629271, −1.09106334559907171303307100883, 0.01472861822061954669922241913, 1.44568975035403485188445841738, 2.62984593056930760965431280500, 3.491027901288152807790050188052, 4.48769346720436677613981697511, 4.777873890129954561798539075283, 6.250360661442435884068163830814, 6.98439689296797512672923962497, 7.9486190506517105468127995848, 8.885736146080896783907893156414, 9.49472033501783348945016315307, 10.43066793331984535523583092590, 10.87362064122509530968107978569, 11.78547187996701170395080327429, 13.22783799199736586321043760553, 13.35988826918297348787013168607, 14.47053579985106697145117685091, 15.16932197485302298666803518416, 15.84867464483866169645297687550, 16.59462127625094543659106187769, 17.27051156412845881887671573833, 18.12284741975918704810462017104, 19.149220371748429303656503383654, 19.98525608680409185020903293460, 20.41300051456322913634124472370

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