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Properties

Label 2-624-52.47-c3-0-41
Degree $2$
Conductor $624$
Sign $0.378 - 0.925i$
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 + (−6.10 − 6.10i)5-s + (−23.8 − 23.8i)7-s − 9·9-s + (−33.3 − 33.3i)11-s + (−42.6 + 19.5i)13-s + (−18.3 + 18.3i)15-s + 20.6i·17-s + (−50.6 + 50.6i)19-s + (−71.5 + 71.5i)21-s + 38.2·23-s − 50.4i·25-s + 27i·27-s + 184.·29-s + (124. − 124. i)31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−0.546 − 0.546i)5-s + (−1.28 − 1.28i)7-s − 0.333·9-s + (−0.914 − 0.914i)11-s + (−0.909 + 0.416i)13-s + (−0.315 + 0.315i)15-s + 0.294i·17-s + (−0.611 + 0.611i)19-s + (−0.743 + 0.743i)21-s + 0.346·23-s − 0.403i·25-s + 0.192i·27-s + 1.17·29-s + (0.719 − 0.719i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.378 - 0.925i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.378 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.378 - 0.925i$
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: \((2,\ 624,\ (\ :3/2),\ 0.378 - 0.925i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.05951427388\)
\(L(\frac12)\) \(\approx\) \(0.05951427388\)
\(L(\frac{5}{2})\) 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 \( 1 + (42.6 - 19.5i)T \)
good5 \( 1 + (6.10 + 6.10i)T + 125iT^{2} \)
7 \( 1 + (23.8 + 23.8i)T + 343iT^{2} \)
11 \( 1 + (33.3 + 33.3i)T + 1.33e3iT^{2} \)
17 \( 1 - 20.6iT - 4.91e3T^{2} \)
19 \( 1 + (50.6 - 50.6i)T - 6.85e3iT^{2} \)
23 \( 1 - 38.2T + 1.21e4T^{2} \)
29 \( 1 - 184.T + 2.43e4T^{2} \)
31 \( 1 + (-124. + 124. i)T - 2.97e4iT^{2} \)
37 \( 1 + (-55.3 + 55.3i)T - 5.06e4iT^{2} \)
41 \( 1 + (-39.7 - 39.7i)T + 6.89e4iT^{2} \)
43 \( 1 - 371.T + 7.95e4T^{2} \)
47 \( 1 + (416. + 416. i)T + 1.03e5iT^{2} \)
53 \( 1 - 34.9T + 1.48e5T^{2} \)
59 \( 1 + (54.1 + 54.1i)T + 2.05e5iT^{2} \)
61 \( 1 + 613.T + 2.26e5T^{2} \)
67 \( 1 + (-694. + 694. i)T - 3.00e5iT^{2} \)
71 \( 1 + (-515. + 515. i)T - 3.57e5iT^{2} \)
73 \( 1 + (216. - 216. i)T - 3.89e5iT^{2} \)
79 \( 1 - 143. iT - 4.93e5T^{2} \)
83 \( 1 + (587. - 587. i)T - 5.71e5iT^{2} \)
89 \( 1 + (967. - 967. i)T - 7.04e5iT^{2} \)
97 \( 1 + (174. + 174. i)T + 9.12e5iT^{2} \)
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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

−9.606070291707687050665383366902, −8.378592510293674916255292490698, −7.77833214254829017193754593399, −6.81779308709087353417491257854, −6.08133817246404113577576034941, −4.70427099315003174413683275427, −3.72674516913046692685294568697, −2.61972378087299070475356513807, −0.78381052898393190654013952159, −0.02393784988939507548725344958, 2.67140918096854675277765139288, 2.93633241977256398610128029491, 4.50749297134762011811524222167, 5.37308390084214518620267478332, 6.47241137304580928299969977752, 7.31447529338626543059667165719, 8.390976566827701927615349104161, 9.377847715758294775034744000764, 9.955559537244972055741013415857, 10.75796845935532050604588098446

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