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Properties

Label 2-2720-85.74-c0-0-1
Degree $2$
Conductor $2720$
Sign $0.563 + 0.825i$
Analytic cond. $1.35745$
Root an. cond. $1.16509$
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  + (0.707 − 0.707i)5-s + (0.923 − 0.382i)9-s + (−1.30 − 1.30i)13-s + (0.707 + 0.707i)17-s − 1.00i·25-s + (0.923 + 0.617i)29-s + (−0.216 − 1.08i)37-s + (−0.923 − 1.38i)41-s + (0.382 − 0.923i)45-s + (−0.382 + 0.923i)49-s + (−0.292 + 0.707i)53-s + (−0.324 + 0.216i)61-s − 1.84·65-s + (1.38 + 0.923i)73-s + (0.707 − 0.707i)81-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)5-s + (0.923 − 0.382i)9-s + (−1.30 − 1.30i)13-s + (0.707 + 0.707i)17-s − 1.00i·25-s + (0.923 + 0.617i)29-s + (−0.216 − 1.08i)37-s + (−0.923 − 1.38i)41-s + (0.382 − 0.923i)45-s + (−0.382 + 0.923i)49-s + (−0.292 + 0.707i)53-s + (−0.324 + 0.216i)61-s − 1.84·65-s + (1.38 + 0.923i)73-s + (0.707 − 0.707i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.563 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.563 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2720\)    =    \(2^{5} \cdot 5 \cdot 17\)
Sign: $0.563 + 0.825i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2720} (2369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2720,\ (\ :0),\ 0.563 + 0.825i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.416416031\)
\(L(\frac12)\) \(\approx\) \(1.416416031\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.707 + 0.707i)T \)
17 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (-0.923 + 0.382i)T^{2} \)
7 \( 1 + (0.382 - 0.923i)T^{2} \)
11 \( 1 + (-0.923 - 0.382i)T^{2} \)
13 \( 1 + (1.30 + 1.30i)T + iT^{2} \)
19 \( 1 + (0.707 + 0.707i)T^{2} \)
23 \( 1 + (-0.923 - 0.382i)T^{2} \)
29 \( 1 + (-0.923 - 0.617i)T + (0.382 + 0.923i)T^{2} \)
31 \( 1 + (-0.923 + 0.382i)T^{2} \)
37 \( 1 + (0.216 + 1.08i)T + (-0.923 + 0.382i)T^{2} \)
41 \( 1 + (0.923 + 1.38i)T + (-0.382 + 0.923i)T^{2} \)
43 \( 1 + (-0.707 + 0.707i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \)
59 \( 1 + (-0.707 + 0.707i)T^{2} \)
61 \( 1 + (0.324 - 0.216i)T + (0.382 - 0.923i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (0.923 - 0.382i)T^{2} \)
73 \( 1 + (-1.38 - 0.923i)T + (0.382 + 0.923i)T^{2} \)
79 \( 1 + (-0.923 - 0.382i)T^{2} \)
83 \( 1 + (0.707 + 0.707i)T^{2} \)
89 \( 1 + (-0.541 - 0.541i)T + iT^{2} \)
97 \( 1 + (1.08 - 1.63i)T + (-0.382 - 0.923i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.981246338655968000036735011238, −8.120902547495026440041278313424, −7.44488465144622084208581640822, −6.61335962295340219951583111744, −5.64929415145086432325291815036, −5.11721754562208857653718313507, −4.24493643227508757101450242144, −3.18494261634964738618873481785, −2.08567026394038673891412409414, −0.979706864441323239136964291577, 1.59986103969466442007789296709, 2.42191361111328973581852436707, 3.38356329856191473128101027456, 4.65171329555900484444839257394, 5.03741107360972722581949045486, 6.29961946115498009205340183991, 6.85260494566454495432979807494, 7.43745072568934262566985050730, 8.300098850824772468191190957953, 9.492632656871410790647631709823

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