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Properties

Label 2-624-52.47-c3-0-16
Degree $2$
Conductor $624$
Sign $-0.553 - 0.833i$
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  + 3i·3-s + (7.38 + 7.38i)5-s + (15.7 + 15.7i)7-s − 9·9-s + (−5.04 − 5.04i)11-s + (36.1 + 29.8i)13-s + (−22.1 + 22.1i)15-s + 46.0i·17-s + (−69.9 + 69.9i)19-s + (−47.1 + 47.1i)21-s + 114.·23-s − 16.0i·25-s − 27i·27-s + 191.·29-s + (−56.8 + 56.8i)31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.660 + 0.660i)5-s + (0.848 + 0.848i)7-s − 0.333·9-s + (−0.138 − 0.138i)11-s + (0.770 + 0.637i)13-s + (−0.381 + 0.381i)15-s + 0.657i·17-s + (−0.845 + 0.845i)19-s + (−0.489 + 0.489i)21-s + 1.03·23-s − 0.128i·25-s − 0.192i·27-s + 1.22·29-s + (−0.329 + 0.329i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 - 0.833i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.553 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-0.553 - 0.833i$
Analytic conductor: \(36.8171\)
Root analytic conductor: \(6.06771\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (463, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :3/2),\ -0.553 - 0.833i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.395884420\)
\(L(\frac12)\) \(\approx\) \(2.395884420\)
\(L(\frac{5}{2})\) 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 \( 1 - 3iT \)
13 \( 1 + (-36.1 - 29.8i)T \)
good5 \( 1 + (-7.38 - 7.38i)T + 125iT^{2} \)
7 \( 1 + (-15.7 - 15.7i)T + 343iT^{2} \)
11 \( 1 + (5.04 + 5.04i)T + 1.33e3iT^{2} \)
17 \( 1 - 46.0iT - 4.91e3T^{2} \)
19 \( 1 + (69.9 - 69.9i)T - 6.85e3iT^{2} \)
23 \( 1 - 114.T + 1.21e4T^{2} \)
29 \( 1 - 191.T + 2.43e4T^{2} \)
31 \( 1 + (56.8 - 56.8i)T - 2.97e4iT^{2} \)
37 \( 1 + (-159. + 159. i)T - 5.06e4iT^{2} \)
41 \( 1 + (91.6 + 91.6i)T + 6.89e4iT^{2} \)
43 \( 1 + 309.T + 7.95e4T^{2} \)
47 \( 1 + (-104. - 104. i)T + 1.03e5iT^{2} \)
53 \( 1 + 392.T + 1.48e5T^{2} \)
59 \( 1 + (178. + 178. i)T + 2.05e5iT^{2} \)
61 \( 1 - 518.T + 2.26e5T^{2} \)
67 \( 1 + (203. - 203. i)T - 3.00e5iT^{2} \)
71 \( 1 + (-62.6 + 62.6i)T - 3.57e5iT^{2} \)
73 \( 1 + (-174. + 174. i)T - 3.89e5iT^{2} \)
79 \( 1 - 722. iT - 4.93e5T^{2} \)
83 \( 1 + (805. - 805. i)T - 5.71e5iT^{2} \)
89 \( 1 + (-66.0 + 66.0i)T - 7.04e5iT^{2} \)
97 \( 1 + (894. + 894. i)T + 9.12e5iT^{2} \)
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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

−10.62195444894941306865709668872, −9.697236250986422988729366262211, −8.691645095586485736361604437615, −8.211525349246626844863577334680, −6.69031946251170021144058043806, −5.97878931231688274588741961116, −5.05229766257399472307827603087, −3.93872309641981610923917887688, −2.66905423506545591947240128897, −1.63072890309804207226950167795, 0.71713859550421417579982959567, 1.56852098806055637258776876117, 2.95247465726442941217549150645, 4.52104565700168558171887025438, 5.21471956829829767814521117798, 6.36898246981561936797733655716, 7.25656951147166777282391466477, 8.209009841954509642771434755286, 8.861793624578009368101047094233, 9.931869399133539270089358455933

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