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Properties

Label 2-3024-63.41-c1-0-17
Degree $2$
Conductor $3024$
Sign $0.637 - 0.770i$
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  + (−1.11 + 1.92i)5-s + (2.58 − 0.566i)7-s + (−1.51 + 0.876i)11-s + (−3.37 − 1.94i)13-s + 1.78·17-s − 2.73i·19-s + (1.64 + 0.947i)23-s + (0.0196 + 0.0339i)25-s + (7.48 − 4.32i)29-s + (7.18 + 4.14i)31-s + (−1.78 + 5.61i)35-s − 9.91·37-s + (3.42 − 5.93i)41-s + (4.22 + 7.32i)43-s + (1.47 + 2.54i)47-s + ⋯
L(s)  = 1  + (−0.498 + 0.862i)5-s + (0.976 − 0.214i)7-s + (−0.457 + 0.264i)11-s + (−0.935 − 0.540i)13-s + 0.433·17-s − 0.628i·19-s + (0.342 + 0.197i)23-s + (0.00392 + 0.00678i)25-s + (1.39 − 0.802i)29-s + (1.29 + 0.745i)31-s + (−0.301 + 0.949i)35-s − 1.63·37-s + (0.535 − 0.927i)41-s + (0.644 + 1.11i)43-s + (0.214 + 0.371i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.637 - 0.770i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.637 - 0.770i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.705450640\)
\(L(\frac12)\) \(\approx\) \(1.705450640\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.58 + 0.566i)T \)
good5 \( 1 + (1.11 - 1.92i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.51 - 0.876i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.37 + 1.94i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 1.78T + 17T^{2} \)
19 \( 1 + 2.73iT - 19T^{2} \)
23 \( 1 + (-1.64 - 0.947i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-7.48 + 4.32i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-7.18 - 4.14i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 9.91T + 37T^{2} \)
41 \( 1 + (-3.42 + 5.93i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.22 - 7.32i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.47 - 2.54i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 5.45iT - 53T^{2} \)
59 \( 1 + (0.449 - 0.778i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-10.3 + 5.98i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.38 - 9.32i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 14.2iT - 71T^{2} \)
73 \( 1 - 4.71iT - 73T^{2} \)
79 \( 1 + (-4.70 - 8.15i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.326 + 0.565i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 9.70T + 89T^{2} \)
97 \( 1 + (0.0294 - 0.0169i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.574207881544806344099748481672, −8.046243662731283844122503568302, −7.22947598665971501566231247444, −6.92423915613288823814131591206, −5.66072518368051458287384878362, −4.91629985010706767229096101345, −4.22611961959915967633569237877, −3.02733332089264207300665689215, −2.45568809326979860917733279543, −0.975491999701444185207316188473, 0.67809414090898722375073993943, 1.83649538684344486046281843902, 2.88941734580461618240020633665, 4.09323375723277969770298293668, 4.85595987386242300262630603179, 5.23521017364011818680838760880, 6.33924084555480820216342180569, 7.30324666122453767280448158501, 8.024612819145887192625196142504, 8.496956432525182863058635685433

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