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Properties

Label 1-2340-2340.1267-r1-0-0
Degree $1$
Conductor $2340$
Sign $0.760 - 0.649i$
Analytic cond. $251.467$
Root an. cond. $251.467$
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  + 7-s + (0.866 + 0.5i)11-s + (0.866 + 0.5i)17-s + (0.866 + 0.5i)19-s − i·23-s + (0.5 − 0.866i)29-s + (−0.866 − 0.5i)31-s + (0.5 + 0.866i)37-s − i·41-s − i·43-s + (−0.5 − 0.866i)47-s + 49-s − i·53-s + (0.866 − 0.5i)59-s + 61-s + ⋯
L(s)  = 1  + 7-s + (0.866 + 0.5i)11-s + (0.866 + 0.5i)17-s + (0.866 + 0.5i)19-s − i·23-s + (0.5 − 0.866i)29-s + (−0.866 − 0.5i)31-s + (0.5 + 0.866i)37-s − i·41-s − i·43-s + (−0.5 − 0.866i)47-s + 49-s − i·53-s + (0.866 − 0.5i)59-s + 61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.760 - 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.760 - 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.760 - 0.649i$
Analytic conductor: \(251.467\)
Root analytic conductor: \(251.467\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (1267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2340,\ (1:\ ),\ 0.760 - 0.649i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.972834525 - 1.096587146i\)
\(L(\frac12)\) \(\approx\) \(2.972834525 - 1.096587146i\)
\(L(1)\) \(\approx\) \(1.391483104 - 0.09027430833i\)
\(L(1)\) \(\approx\) \(1.391483104 - 0.09027430833i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good7 \( 1 + T \)
11 \( 1 + (0.866 + 0.5i)T \)
17 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 - iT \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (-0.866 - 0.5i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 - iT \)
43 \( 1 - iT \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 - iT \)
59 \( 1 + (0.866 - 0.5i)T \)
61 \( 1 + T \)
67 \( 1 - T \)
71 \( 1 + (-0.866 - 0.5i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
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

−19.70677422852482939308209694935, −18.6946805974063311492854484418, −17.96880115659364577553595811771, −17.53593372195239933730371474020, −16.516416103412172206973318379773, −16.10937246468987776073382116559, −15.05337874847466976366306328423, −14.32486736701485329710350496333, −14.00668161699663144113357347797, −13.008642425572808228743530013044, −12.105036036340231550742391165, −11.41012083081421643244816909226, −11.028227563865332002283524959784, −9.871881529293611543419165718062, −9.222428258644342252429535335729, −8.46016044141302310830547243698, −7.59548183613621542212685811336, −7.046336167013834759319039885933, −5.9000437924797722760314565179, −5.27162313695310972464975027004, −4.45652793335085214978824358429, −3.50334071695929264960473249068, −2.736349717828520947302303221093, −1.4052791223128618588509669754, −1.026492316783720534557546727945, 0.59466182884898930423927029186, 1.53490947792603784751176402154, 2.241202896334348663002304634606, 3.50041503577547339415457341413, 4.18420613114422623277310084354, 5.05703682222932960746266976256, 5.79850599530541381913174680884, 6.73286557134603255281318229688, 7.557939035817654683318432854130, 8.2341228987058374995625403326, 8.96966653473090582374866273451, 9.94440732333527637587816091949, 10.45481844807397566362822566051, 11.630270344364281759714084342224, 11.845795750994615809580801437638, 12.75160369377677929101615802901, 13.69047004331600096236084848080, 14.559315682376157404712658006176, 14.71264828425058985888850175261, 15.72432722413609380785624623903, 16.67407620286304671209687602537, 17.13818527930965635694191820775, 17.91788455126837105150005791619, 18.58413864857205124271374480256, 19.28493990203696637992660714934

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