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Properties

Label 2-3024-63.58-c1-0-21
Degree $2$
Conductor $3024$
Sign $0.753 + 0.656i$
Analytic cond. $24.1467$
Root an. cond. $4.91393$
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  + (−0.381 − 0.661i)5-s + (−2.62 − 0.297i)7-s + (−3.01 + 5.22i)11-s + (−1.26 + 2.18i)13-s + (−1.94 − 3.36i)17-s + (2.13 − 3.69i)19-s + (0.732 + 1.26i)23-s + (2.20 − 3.82i)25-s + (3.00 + 5.20i)29-s − 6.56·31-s + (0.806 + 1.85i)35-s + (4.82 − 8.35i)37-s + (2.24 − 3.89i)41-s + (2.13 + 3.69i)43-s − 6.77·47-s + ⋯
L(s)  = 1  + (−0.170 − 0.295i)5-s + (−0.993 − 0.112i)7-s + (−0.909 + 1.57i)11-s + (−0.349 + 0.605i)13-s + (−0.471 − 0.816i)17-s + (0.489 − 0.848i)19-s + (0.152 + 0.264i)23-s + (0.441 − 0.764i)25-s + (0.558 + 0.967i)29-s − 1.17·31-s + (0.136 + 0.313i)35-s + (0.793 − 1.37i)37-s + (0.351 − 0.608i)41-s + (0.325 + 0.563i)43-s − 0.988·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.753 + 0.656i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.753 + 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.753 + 0.656i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.753 + 0.656i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.076419487\)
\(L(\frac12)\) \(\approx\) \(1.076419487\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (2.62 + 0.297i)T \)
good5 \( 1 + (0.381 + 0.661i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.01 - 5.22i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.26 - 2.18i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.94 + 3.36i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.13 + 3.69i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.732 - 1.26i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.00 - 5.20i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.56T + 31T^{2} \)
37 \( 1 + (-4.82 + 8.35i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.24 + 3.89i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.13 - 3.69i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 6.77T + 47T^{2} \)
53 \( 1 + (-0.265 - 0.459i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 11.1T + 59T^{2} \)
61 \( 1 - 8.38T + 61T^{2} \)
67 \( 1 - 1.92T + 67T^{2} \)
71 \( 1 - 9.90T + 71T^{2} \)
73 \( 1 + (2.13 + 3.69i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 7.40T + 79T^{2} \)
83 \( 1 + (8.05 + 13.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.76 - 3.05i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.33 - 4.04i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.856940951141240925756391872954, −7.66219654526423560792275682105, −7.11526213183927253542093390468, −6.64511175758136985779366609654, −5.37603114754225948001601322803, −4.80218834468094826963315859659, −4.00396205402680973687647988477, −2.83136176687568440488107562125, −2.13300112897170189056774227405, −0.47603889231972868135274661697, 0.77528713583386275087757215681, 2.45424483927030933804095197761, 3.21996367900665440916888631812, 3.80845910479803758450137126889, 5.14981548920946933780433283987, 5.83432255526525791577764094361, 6.42377900362879603546304881350, 7.33338660932100268114829477512, 8.205391875399938146045294533487, 8.578524039160963251261911461145

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