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Properties

Label 2-624-12.11-c3-0-10
Degree $2$
Conductor $624$
Sign $-0.439 - 0.898i$
Analytic cond. $36.8171$
Root an. cond. $6.06771$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.90 + 4.31i)3-s − 9.14i·5-s + 18.5i·7-s + (−10.1 − 25.0i)9-s + 8.04·11-s + 13·13-s + (39.4 + 26.5i)15-s − 37.3i·17-s + 42.4i·19-s + (−79.8 − 53.7i)21-s + 19.9·23-s + 41.4·25-s + (137. + 28.7i)27-s − 9.94i·29-s + 30.2i·31-s + ⋯
L(s)  = 1  + (−0.558 + 0.829i)3-s − 0.817i·5-s + 0.999i·7-s + (−0.376 − 0.926i)9-s + 0.220·11-s + 0.277·13-s + (0.678 + 0.456i)15-s − 0.533i·17-s + 0.512i·19-s + (−0.829 − 0.558i)21-s + 0.180·23-s + 0.331·25-s + (0.978 + 0.204i)27-s − 0.0637i·29-s + 0.175i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.439 - 0.898i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.439 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-0.439 - 0.898i$
Analytic conductor: \(36.8171\)
Root analytic conductor: \(6.06771\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :3/2),\ -0.439 - 0.898i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.160238166\)
\(L(\frac12)\) \(\approx\) \(1.160238166\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.90 - 4.31i)T \)
13 \( 1 - 13T \)
good5 \( 1 + 9.14iT - 125T^{2} \)
7 \( 1 - 18.5iT - 343T^{2} \)
11 \( 1 - 8.04T + 1.33e3T^{2} \)
17 \( 1 + 37.3iT - 4.91e3T^{2} \)
19 \( 1 - 42.4iT - 6.85e3T^{2} \)
23 \( 1 - 19.9T + 1.21e4T^{2} \)
29 \( 1 + 9.94iT - 2.43e4T^{2} \)
31 \( 1 - 30.2iT - 2.97e4T^{2} \)
37 \( 1 + 116.T + 5.06e4T^{2} \)
41 \( 1 - 308. iT - 6.89e4T^{2} \)
43 \( 1 - 26.8iT - 7.95e4T^{2} \)
47 \( 1 - 111.T + 1.03e5T^{2} \)
53 \( 1 - 331. iT - 1.48e5T^{2} \)
59 \( 1 + 327.T + 2.05e5T^{2} \)
61 \( 1 + 130.T + 2.26e5T^{2} \)
67 \( 1 - 345. iT - 3.00e5T^{2} \)
71 \( 1 - 55.5T + 3.57e5T^{2} \)
73 \( 1 + 270.T + 3.89e5T^{2} \)
79 \( 1 - 927. iT - 4.93e5T^{2} \)
83 \( 1 + 173.T + 5.71e5T^{2} \)
89 \( 1 - 1.52e3iT - 7.04e5T^{2} \)
97 \( 1 + 1.01e3T + 9.12e5T^{2} \)
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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

−10.48130281610123600193654412746, −9.453280424224461018276121679727, −8.985376492342022315950312322078, −8.139261173774417972364486829393, −6.67595396095378206141822741611, −5.70661506116104410824116605555, −5.06030688910624284113951630829, −4.10756486490978924777472967184, −2.84392485887496582117151627261, −1.14443494799873273597077703646, 0.41273755114121763843978639815, 1.69793995267680090785912957429, 3.06387893196118168656017378808, 4.27808185527644744476297757137, 5.53371738614102650524352557028, 6.58925333463244069304550624877, 7.06748161449370062521840605512, 7.893026047228527422480343081186, 8.973421075694714396965664060104, 10.36474656760136616932098669685

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