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Properties

Label 2-5220-145.144-c1-0-51
Degree $2$
Conductor $5220$
Sign $0.996 - 0.0876i$
Analytic cond. $41.6819$
Root an. cond. $6.45615$
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  + (2.10 + 0.754i)5-s + 1.11i·7-s + 0.423i·11-s − 3.69i·13-s + 6.28·17-s + 3.85i·19-s − 3.79i·23-s + (3.86 + 3.17i)25-s + (4.89 − 2.25i)29-s − 7.17i·31-s + (−0.838 + 2.33i)35-s − 5.33·37-s + 6.08i·41-s + 7.16·43-s − 8.32·47-s + ⋯
L(s)  = 1  + (0.941 + 0.337i)5-s + 0.419i·7-s + 0.127i·11-s − 1.02i·13-s + 1.52·17-s + 0.884i·19-s − 0.790i·23-s + (0.772 + 0.635i)25-s + (0.908 − 0.418i)29-s − 1.28i·31-s + (−0.141 + 0.395i)35-s − 0.877·37-s + 0.950i·41-s + 1.09·43-s − 1.21·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0876i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5220\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 29\)
Sign: $0.996 - 0.0876i$
Analytic conductor: \(41.6819\)
Root analytic conductor: \(6.45615\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5220} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5220,\ (\ :1/2),\ 0.996 - 0.0876i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.609349557\)
\(L(\frac12)\) \(\approx\) \(2.609349557\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.10 - 0.754i)T \)
29 \( 1 + (-4.89 + 2.25i)T \)
good7 \( 1 - 1.11iT - 7T^{2} \)
11 \( 1 - 0.423iT - 11T^{2} \)
13 \( 1 + 3.69iT - 13T^{2} \)
17 \( 1 - 6.28T + 17T^{2} \)
19 \( 1 - 3.85iT - 19T^{2} \)
23 \( 1 + 3.79iT - 23T^{2} \)
31 \( 1 + 7.17iT - 31T^{2} \)
37 \( 1 + 5.33T + 37T^{2} \)
41 \( 1 - 6.08iT - 41T^{2} \)
43 \( 1 - 7.16T + 43T^{2} \)
47 \( 1 + 8.32T + 47T^{2} \)
53 \( 1 + 7.82iT - 53T^{2} \)
59 \( 1 + 1.94T + 59T^{2} \)
61 \( 1 - 12.8iT - 61T^{2} \)
67 \( 1 + 14.8iT - 67T^{2} \)
71 \( 1 - 3.62T + 71T^{2} \)
73 \( 1 - 1.15T + 73T^{2} \)
79 \( 1 - 3.85iT - 79T^{2} \)
83 \( 1 + 6.56iT - 83T^{2} \)
89 \( 1 + 11.4iT - 89T^{2} \)
97 \( 1 - 0.415T + 97T^{2} \)
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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

−8.083288129618822472746485956517, −7.66886740471743536678728398708, −6.62091274991513879728919819642, −5.93690238404702933170577495583, −5.53021612858847755121669015934, −4.68138497941698840754776358078, −3.51608409864221864872703950309, −2.83375462725186491103520622587, −1.98277878596190805451522225653, −0.878186869662910901624670873827, 0.964195743213655488433613311224, 1.73134738001626759340979318171, 2.82065675783613976838881816466, 3.65963370365861631914379523980, 4.66976671875692359153524942293, 5.27530056639087176180903068799, 5.97990148115754995514273495260, 6.85800603167105139643281447240, 7.28520959245692278422188273576, 8.332133466067022974665199878201

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