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Properties

Label 1-2720-2720.3-r1-0-0
Degree $1$
Conductor $2720$
Sign $-0.170 - 0.985i$
Analytic cond. $292.304$
Root an. cond. $292.304$
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  + (0.923 − 0.382i)3-s + (−0.382 − 0.923i)7-s + (0.707 − 0.707i)9-s + (0.382 − 0.923i)11-s + (0.707 + 0.707i)13-s − 19-s + (−0.707 − 0.707i)21-s + (0.923 − 0.382i)23-s + (0.382 − 0.923i)27-s + (−0.923 + 0.382i)29-s + (0.923 + 0.382i)31-s − i·33-s + (0.382 + 0.923i)37-s + (0.923 + 0.382i)39-s + (0.923 − 0.382i)41-s + ⋯
L(s)  = 1  + (0.923 − 0.382i)3-s + (−0.382 − 0.923i)7-s + (0.707 − 0.707i)9-s + (0.382 − 0.923i)11-s + (0.707 + 0.707i)13-s − 19-s + (−0.707 − 0.707i)21-s + (0.923 − 0.382i)23-s + (0.382 − 0.923i)27-s + (−0.923 + 0.382i)29-s + (0.923 + 0.382i)31-s − i·33-s + (0.382 + 0.923i)37-s + (0.923 + 0.382i)39-s + (0.923 − 0.382i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.170 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.170 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2720\)    =    \(2^{5} \cdot 5 \cdot 17\)
Sign: $-0.170 - 0.985i$
Analytic conductor: \(292.304\)
Root analytic conductor: \(292.304\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2720} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2720,\ (1:\ ),\ -0.170 - 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.368330834 - 2.812327108i\)
\(L(\frac12)\) \(\approx\) \(2.368330834 - 2.812327108i\)
\(L(1)\) \(\approx\) \(1.480573409 - 0.5568499115i\)
\(L(1)\) \(\approx\) \(1.480573409 - 0.5568499115i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
17 \( 1 \)
good3 \( 1 + (0.923 - 0.382i)T \)
7 \( 1 + (-0.382 - 0.923i)T \)
11 \( 1 + (0.382 - 0.923i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
19 \( 1 - T \)
23 \( 1 + (0.923 - 0.382i)T \)
29 \( 1 + (-0.923 + 0.382i)T \)
31 \( 1 + (0.923 + 0.382i)T \)
37 \( 1 + (0.382 + 0.923i)T \)
41 \( 1 + (0.923 - 0.382i)T \)
43 \( 1 - iT \)
47 \( 1 + T \)
53 \( 1 + T \)
59 \( 1 + T \)
61 \( 1 + (-0.382 + 0.923i)T \)
67 \( 1 + (0.707 - 0.707i)T \)
71 \( 1 + (-0.382 + 0.923i)T \)
73 \( 1 + (0.382 - 0.923i)T \)
79 \( 1 + (0.923 - 0.382i)T \)
83 \( 1 - iT \)
89 \( 1 + T \)
97 \( 1 + (-0.923 - 0.382i)T \)
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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

−19.33657467103017696528463366081, −18.7609091412533274322407024449, −17.97854525619185184066978811974, −17.193221677690522155596307862722, −16.27988705013872574579813443035, −15.55381592008602988540170966084, −15.03020659830548173271498368092, −14.64373794505827620933709864793, −13.51735948155833390761475595586, −12.932895929933189270907748284018, −12.40843136961042352805997320687, −11.314506533755260240479956870659, −10.60745643263649111912007252010, −9.64003934963116255621559389227, −9.28809351733757822580381058499, −8.47681432890498246991470023286, −7.83308068434890627688399814407, −6.93113052216514874463897634105, −6.05381799964901960610638983679, −5.218342500877592470415569758187, −4.26375403553991041679352524589, −3.62607315533788719749224698043, −2.61758800528795557747051675068, −2.14740612610844554428738254019, −0.97971599312123726406935845903, 0.58545995174667742174486136134, 1.234579152396893561345862770917, 2.26950846264243750002581673707, 3.213864611099790459936888912406, 3.86550567297754579311707837566, 4.47294992215792349146154154976, 5.87159456032996779509137201615, 6.65907719432436413449345621627, 7.104193335652272281871560275815, 8.09285776804756155801102845299, 8.794951379507055985159093563892, 9.244368653533273680792491188903, 10.32091484202697239456080193975, 10.88706761961506049890460128341, 11.79693336961710377680974305004, 12.71519287659197740524961324644, 13.433380339065771221804863714923, 13.77722907005182213705323512200, 14.54317708940809414558860763198, 15.23317372314251279121848508229, 16.15682796093655786181527574364, 16.74757271653745536312878609968, 17.42497093022797048916190424736, 18.49650955884367216575483874128, 19.060770360285843998879789070763

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