Properties

Degree $2$
Conductor $3024$
Sign $0.968 - 0.247i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.59 − 2.76i)5-s + (1.66 + 2.05i)7-s + (1.14 + 1.97i)11-s + (−0.675 − 1.16i)13-s + (−2.21 + 3.83i)17-s + (3.69 + 6.39i)19-s + (3.23 − 5.60i)23-s + (−2.60 − 4.51i)25-s + (1.06 − 1.83i)29-s + 0.632·31-s + (8.34 − 1.32i)35-s + (1.92 + 3.34i)37-s + (5.05 + 8.74i)41-s + (−4.24 + 7.35i)43-s + 6.53·47-s + ⋯
L(s)  = 1  + (0.714 − 1.23i)5-s + (0.629 + 0.776i)7-s + (0.344 + 0.596i)11-s + (−0.187 − 0.324i)13-s + (−0.537 + 0.930i)17-s + (0.847 + 1.46i)19-s + (0.674 − 1.16i)23-s + (−0.520 − 0.902i)25-s + (0.197 − 0.341i)29-s + 0.113·31-s + (1.41 − 0.224i)35-s + (0.317 + 0.549i)37-s + (0.788 + 1.36i)41-s + (−0.647 + 1.12i)43-s + 0.952·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.247i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.968 - 0.247i$
Motivic weight: \(1\)
Character: $\chi_{3024} (2881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.968 - 0.247i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.380762760\)
\(L(\frac12)\) \(\approx\) \(2.380762760\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.66 - 2.05i)T \)
good5 \( 1 + (-1.59 + 2.76i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.14 - 1.97i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.675 + 1.16i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.21 - 3.83i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.69 - 6.39i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.23 + 5.60i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.06 + 1.83i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.632T + 31T^{2} \)
37 \( 1 + (-1.92 - 3.34i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.05 - 8.74i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.24 - 7.35i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 6.53T + 47T^{2} \)
53 \( 1 + (2.39 - 4.15i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 6.20T + 59T^{2} \)
61 \( 1 + 8.91T + 61T^{2} \)
67 \( 1 - 3.01T + 67T^{2} \)
71 \( 1 + 15.3T + 71T^{2} \)
73 \( 1 + (-4.36 + 7.56i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 1.87T + 79T^{2} \)
83 \( 1 + (3.00 - 5.19i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.65 + 4.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.44 + 12.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.588711714965075432015175247893, −8.314014199977195729044101225161, −7.39378648954059393275598640356, −6.09996068000770316002305789051, −5.83766361352547001333171770923, −4.72040089514807669682878492286, −4.47298510517335242503747663678, −2.97351830479727799730644894433, −1.86914528254694528369392380366, −1.20511219694422419821540126495, 0.845446010663431617011472203577, 2.15610401503750622141429888147, 2.97382882287461549045494947899, 3.84491218928161140681226517179, 4.93454316586519496271793802700, 5.59980068258505305751510600870, 6.70607176232630503781685696082, 7.09007308769634336634747429178, 7.64028888852308979335869473568, 8.983525924437525868996386187474

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