Properties

Label 2-624-39.2-c1-0-16
Degree $2$
Conductor $624$
Sign $0.963 + 0.268i$
Analytic cond. $4.98266$
Root an. cond. $2.23218$
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  + (1.28 − 1.16i)3-s + (1.69 + 1.69i)5-s + (1.36 − 0.366i)7-s + (0.292 − 2.98i)9-s + (1.69 + 0.453i)11-s + (−1.59 + 3.23i)13-s + (4.14 + 0.202i)15-s + (1.07 − 1.85i)17-s + (0.267 + i)19-s + (1.32 − 2.05i)21-s + 0.732i·25-s + (−3.09 − 4.17i)27-s + (4.79 − 2.76i)29-s + (−4.46 + 4.46i)31-s + (2.69 − 1.38i)33-s + ⋯
L(s)  = 1  + (0.740 − 0.671i)3-s + (0.757 + 0.757i)5-s + (0.516 − 0.138i)7-s + (0.0975 − 0.995i)9-s + (0.510 + 0.136i)11-s + (−0.443 + 0.896i)13-s + (1.06 + 0.0523i)15-s + (0.260 − 0.450i)17-s + (0.0614 + 0.229i)19-s + (0.289 − 0.449i)21-s + 0.146i·25-s + (−0.596 − 0.802i)27-s + (0.889 − 0.513i)29-s + (−0.801 + 0.801i)31-s + (0.470 − 0.241i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 + 0.268i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.963 + 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.963 + 0.268i$
Analytic conductor: \(4.98266\)
Root analytic conductor: \(2.23218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :1/2),\ 0.963 + 0.268i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.23966 - 0.306549i\)
\(L(\frac12)\) \(\approx\) \(2.23966 - 0.306549i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.28 + 1.16i)T \)
13 \( 1 + (1.59 - 3.23i)T \)
good5 \( 1 + (-1.69 - 1.69i)T + 5iT^{2} \)
7 \( 1 + (-1.36 + 0.366i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-1.69 - 0.453i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.07 + 1.85i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.267 - i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.79 + 2.76i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.46 - 4.46i)T - 31iT^{2} \)
37 \( 1 + (1.76 - 6.59i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (0.166 - 0.619i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (7.09 + 4.09i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.77 + 6.77i)T - 47iT^{2} \)
53 \( 1 + 4.62iT - 53T^{2} \)
59 \( 1 + (1.23 + 4.62i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.46 - 2.26i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (4.62 - 1.23i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (6.09 + 6.09i)T + 73iT^{2} \)
79 \( 1 + 2T + 79T^{2} \)
83 \( 1 + (-1.23 - 1.23i)T + 83iT^{2} \)
89 \( 1 + (9.70 + 2.60i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-3.36 - 12.5i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.38106697208301844202661360169, −9.675926689006692865745692247545, −8.830356615641540883239295859192, −7.897034025897756341533857261323, −6.87136866341011276503099578471, −6.48154660441220887355035311408, −5.06536397895484068475810801163, −3.72044108469972944741742403507, −2.52865162578355375189866279071, −1.57482619999457212044462351694, 1.55481508596812959132166508044, 2.83573064122903657920032121676, 4.11368456079074009281044834302, 5.11077643778160358064101348762, 5.79620636917876864939049069327, 7.33121631600401117175524381193, 8.278075875193002467978717603509, 8.946960074085322691248004718331, 9.676704957507271011621732163033, 10.43759439067414548438804868406

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