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Properties

Label 2-2520-105.104-c1-0-15
Degree $2$
Conductor $2520$
Sign $0.452 - 0.891i$
Analytic cond. $20.1223$
Root an. cond. $4.48578$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.727 − 2.11i)5-s + (1.21 + 2.35i)7-s + 2.91i·11-s + 0.380·13-s + 7.43i·17-s − 4.51i·19-s + 2.63·23-s + (−3.94 − 3.07i)25-s + 8.44i·29-s − 5.79i·31-s + (5.85 − 0.858i)35-s + 9.46i·37-s − 0.570·41-s + 2.65i·43-s + 8.79i·47-s + ⋯
L(s)  = 1  + (0.325 − 0.945i)5-s + (0.459 + 0.888i)7-s + 0.878i·11-s + 0.105·13-s + 1.80i·17-s − 1.03i·19-s + 0.548·23-s + (−0.788 − 0.615i)25-s + 1.56i·29-s − 1.04i·31-s + (0.989 − 0.145i)35-s + 1.55i·37-s − 0.0891·41-s + 0.405i·43-s + 1.28i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.452 - 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.452 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.452 - 0.891i$
Analytic conductor: \(20.1223\)
Root analytic conductor: \(4.48578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :1/2),\ 0.452 - 0.891i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.784456682\)
\(L(\frac12)\) \(\approx\) \(1.784456682\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-0.727 + 2.11i)T \)
7 \( 1 + (-1.21 - 2.35i)T \)
good11 \( 1 - 2.91iT - 11T^{2} \)
13 \( 1 - 0.380T + 13T^{2} \)
17 \( 1 - 7.43iT - 17T^{2} \)
19 \( 1 + 4.51iT - 19T^{2} \)
23 \( 1 - 2.63T + 23T^{2} \)
29 \( 1 - 8.44iT - 29T^{2} \)
31 \( 1 + 5.79iT - 31T^{2} \)
37 \( 1 - 9.46iT - 37T^{2} \)
41 \( 1 + 0.570T + 41T^{2} \)
43 \( 1 - 2.65iT - 43T^{2} \)
47 \( 1 - 8.79iT - 47T^{2} \)
53 \( 1 + 14.0T + 53T^{2} \)
59 \( 1 + 6.55T + 59T^{2} \)
61 \( 1 - 2.66iT - 61T^{2} \)
67 \( 1 + 8.87iT - 67T^{2} \)
71 \( 1 + 13.6iT - 71T^{2} \)
73 \( 1 - 6.55T + 73T^{2} \)
79 \( 1 - 6.74T + 79T^{2} \)
83 \( 1 - 15.5iT - 83T^{2} \)
89 \( 1 - 17.4T + 89T^{2} \)
97 \( 1 - 6.88T + 97T^{2} \)
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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

−9.179659061992512597136536511424, −8.274020407749269444948994682681, −7.80039701099587126408896476541, −6.53876711018176508464915693239, −5.97352474240851429644988082734, −4.83500783941956067361321598582, −4.70292104773126675741338343624, −3.30645058310339848767209173922, −2.08733327659210346730964068520, −1.35849446257897876194309011984, 0.60883760258740653295677667661, 1.99281819601589284106615812255, 3.08936673695780459729468498957, 3.77273383066222336014004333335, 4.86908308720662145468526197266, 5.72154350520903877638815631854, 6.53789710117877271571353362552, 7.31128800932051315267189164717, 7.80514991588820602461407988433, 8.784591768814762018981476134361

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