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Properties

Label 2-3024-9.4-c1-0-18
Degree $2$
Conductor $3024$
Sign $0.995 + 0.0952i$
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  + (1.21 − 2.10i)5-s + (0.5 + 0.866i)7-s + (−0.379 − 0.657i)11-s + (−1.11 + 1.92i)13-s + 7.04·17-s + 1.37·19-s + (−3.51 + 6.09i)23-s + (−0.467 − 0.810i)25-s + (0.418 + 0.724i)29-s + (−0.265 + 0.459i)31-s + 2.43·35-s + 4.53·37-s + (4.42 − 7.67i)41-s + (3.70 + 6.41i)43-s + (3.39 + 5.88i)47-s + ⋯
L(s)  = 1  + (0.544 − 0.943i)5-s + (0.188 + 0.327i)7-s + (−0.114 − 0.198i)11-s + (−0.309 + 0.535i)13-s + 1.70·17-s + 0.314·19-s + (−0.733 + 1.27i)23-s + (−0.0935 − 0.162i)25-s + (0.0776 + 0.134i)29-s + (−0.0476 + 0.0825i)31-s + 0.411·35-s + 0.744·37-s + (0.691 − 1.19i)41-s + (0.564 + 0.977i)43-s + (0.495 + 0.858i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.995 + 0.0952i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.995 + 0.0952i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.198145589\)
\(L(\frac12)\) \(\approx\) \(2.198145589\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-1.21 + 2.10i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.379 + 0.657i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.11 - 1.92i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 7.04T + 17T^{2} \)
19 \( 1 - 1.37T + 19T^{2} \)
23 \( 1 + (3.51 - 6.09i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.418 - 0.724i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.265 - 0.459i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.53T + 37T^{2} \)
41 \( 1 + (-4.42 + 7.67i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.70 - 6.41i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.39 - 5.88i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 0.607T + 53T^{2} \)
59 \( 1 + (-0.581 + 1.00i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.85 + 10.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.152 - 0.264i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.6T + 71T^{2} \)
73 \( 1 + 10.1T + 73T^{2} \)
79 \( 1 + (-7.62 - 13.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.18 + 14.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 9.63T + 89T^{2} \)
97 \( 1 + (-5.46 - 9.46i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.833184002475779957139747405061, −7.86962820578955791323345181913, −7.48484551857021305627657863446, −6.19731594629744250636611138306, −5.57561262369970754354230505807, −5.04107795774477190066375596132, −4.04538463647080104641328131680, −3.05290002760228532279504759942, −1.87789860000015574753206642959, −1.00879396067688188540612138717, 0.882015472878540643361436323280, 2.26724223736583658683173302202, 2.97119707682047177384403464077, 3.92700468965628466727728483596, 4.93533389988374848383276377451, 5.82606206802478463325160660933, 6.37283196720717914025015736697, 7.39652853604699045057106499087, 7.75351736404834732936683198692, 8.688018558119374189817139657619

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