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Properties

Label 1-3360-3360.2243-r0-0-0
Degree $1$
Conductor $3360$
Sign $0.773 + 0.633i$
Analytic cond. $15.6037$
Root an. cond. $15.6037$
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  + (−0.965 − 0.258i)11-s + (−0.707 + 0.707i)13-s + (0.866 − 0.5i)17-s + (−0.965 + 0.258i)19-s + (0.5 − 0.866i)23-s + (0.707 + 0.707i)29-s + (−0.5 − 0.866i)31-s + (−0.965 + 0.258i)37-s + i·41-s + (−0.707 − 0.707i)43-s + (0.866 + 0.5i)47-s + (−0.258 + 0.965i)53-s + (0.965 + 0.258i)59-s + (0.965 − 0.258i)61-s + (0.965 + 0.258i)67-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)11-s + (−0.707 + 0.707i)13-s + (0.866 − 0.5i)17-s + (−0.965 + 0.258i)19-s + (0.5 − 0.866i)23-s + (0.707 + 0.707i)29-s + (−0.5 − 0.866i)31-s + (−0.965 + 0.258i)37-s + i·41-s + (−0.707 − 0.707i)43-s + (0.866 + 0.5i)47-s + (−0.258 + 0.965i)53-s + (0.965 + 0.258i)59-s + (0.965 − 0.258i)61-s + (0.965 + 0.258i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.773 + 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.773 + 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3360\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.773 + 0.633i$
Analytic conductor: \(15.6037\)
Root analytic conductor: \(15.6037\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3360} (2243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3360,\ (0:\ ),\ 0.773 + 0.633i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.155984575 + 0.4127053692i\)
\(L(\frac12)\) \(\approx\) \(1.155984575 + 0.4127053692i\)
\(L(1)\) \(\approx\) \(0.9405037796 + 0.04420206716i\)
\(L(1)\) \(\approx\) \(0.9405037796 + 0.04420206716i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + (-0.965 - 0.258i)T \)
13 \( 1 + (-0.707 + 0.707i)T \)
17 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (-0.965 + 0.258i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.707 + 0.707i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.965 + 0.258i)T \)
41 \( 1 + iT \)
43 \( 1 + (-0.707 - 0.707i)T \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + (-0.258 + 0.965i)T \)
59 \( 1 + (0.965 + 0.258i)T \)
61 \( 1 + (0.965 - 0.258i)T \)
67 \( 1 + (0.965 + 0.258i)T \)
71 \( 1 - iT \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (-0.707 + 0.707i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + iT \)
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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

−18.90746466204518726430600819291, −17.91826636679429836277234943270, −17.42513453115557977051966054018, −16.82840651224769143749770036292, −15.8004658827692204644233531938, −15.40055376420505881825502707594, −14.63957195060623409015176427313, −13.96353276784156530574877166136, −12.90553577756742751762483234030, −12.730796618169974590108135913640, −11.81856491518261678480310734144, −10.95360600267623188268947945804, −10.19479190448733943255242324583, −9.86408035393394292639197107143, −8.67003856313331954863979097564, −8.16079939698510649709585598285, −7.32415926735607095354266262393, −6.73502398578718166032854125047, −5.443956013820689803012291271953, −5.33058506711478040639074055099, −4.21006062216890098325528793666, −3.32867838733172830068596145543, −2.549915574221431222862066065798, −1.722136639932411868647852242884, −0.48303396634946476987378567085, 0.75321130609851557291428604773, 1.97397901257805389290141618774, 2.67042029388025679468019045138, 3.50682045423769349542509817682, 4.549274280090386685913712352881, 5.096180040678028418538943354040, 5.962378039319528345476588669628, 6.820556149692763838891832509993, 7.48240554963923197939764947416, 8.29711674834904619885157361283, 8.94080898053327293649482380484, 9.86320383088809653483693208690, 10.435180765671247501462244765005, 11.14437801050190307147778531621, 12.04330045762314221448167426473, 12.57678406958008630612074925696, 13.329195388435574995306031147915, 14.1377994535247006026846224663, 14.70221007148856203162300909600, 15.42451100031229907704441284109, 16.30425150050788186473082894427, 16.737270530935000028053029567650, 17.453064389219741077109996902842, 18.44724657243491501589248106106, 18.80223899836816740544269343259

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