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Properties

Label 2-624-13.3-c3-0-19
Degree $2$
Conductor $624$
Sign $0.477 - 0.878i$
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  + (1.5 + 2.59i)3-s − 9·5-s + (1 − 1.73i)7-s + (−4.5 + 7.79i)9-s + (15 + 25.9i)11-s + (32.5 − 33.7i)13-s + (−13.5 − 23.3i)15-s + (55.5 − 96.1i)17-s + (−23 + 39.8i)19-s + 6·21-s + (−3 − 5.19i)23-s − 44·25-s − 27·27-s + (52.5 + 90.9i)29-s + 100·31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s − 0.804·5-s + (0.0539 − 0.0935i)7-s + (−0.166 + 0.288i)9-s + (0.411 + 0.712i)11-s + (0.693 − 0.720i)13-s + (−0.232 − 0.402i)15-s + (0.791 − 1.37i)17-s + (−0.277 + 0.481i)19-s + 0.0623·21-s + (−0.0271 − 0.0471i)23-s − 0.351·25-s − 0.192·27-s + (0.336 + 0.582i)29-s + 0.579·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 - 0.878i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.477 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.477 - 0.878i$
Analytic conductor: \(36.8171\)
Root analytic conductor: \(6.06771\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :3/2),\ 0.477 - 0.878i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.855679336\)
\(L(\frac12)\) \(\approx\) \(1.855679336\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.5 - 2.59i)T \)
13 \( 1 + (-32.5 + 33.7i)T \)
good5 \( 1 + 9T + 125T^{2} \)
7 \( 1 + (-1 + 1.73i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-15 - 25.9i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-55.5 + 96.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (23 - 39.8i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-52.5 - 90.9i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 100T + 2.97e4T^{2} \)
37 \( 1 + (8.5 + 14.7i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-115.5 - 200. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (257 - 445. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 162T + 1.03e5T^{2} \)
53 \( 1 - 639T + 1.48e5T^{2} \)
59 \( 1 + (-300 + 519. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (116.5 - 201. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-463 - 801. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (465 - 805. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 253T + 3.89e5T^{2} \)
79 \( 1 - 1.32e3T + 4.93e5T^{2} \)
83 \( 1 + 810T + 5.71e5T^{2} \)
89 \( 1 + (249 + 431. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (679 - 1.17e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−10.21759280088295748589653924631, −9.608747201184535224727786924457, −8.521507810361313081651397849990, −7.82734748335094717092900164805, −6.96947746243986636099100937777, −5.69286067065807715021971876738, −4.62523374874596951295113004997, −3.76592902164759883703600280283, −2.77018826177645645544112788036, −1.01552289057209752215427353112, 0.66056249898241450156180098984, 1.98306304860140211172678548734, 3.49818213348786110324906019496, 4.11845221358513004580139850647, 5.68448828610927526421410735388, 6.51612800706993630936987751885, 7.47619997275104978454817976580, 8.440931726225022840282961220656, 8.791863087322136956645716701519, 10.12228070641966794824395147730

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