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Properties

Label 2-3024-63.4-c1-0-12
Degree $2$
Conductor $3024$
Sign $-0.296 - 0.954i$
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  + 3·5-s + (0.5 + 2.59i)7-s − 3·11-s + (−2.5 + 4.33i)13-s + (1.5 − 2.59i)17-s + (2.5 + 4.33i)19-s − 3·23-s + 4·25-s + (−1.5 − 2.59i)29-s + (−2 − 3.46i)31-s + (1.5 + 7.79i)35-s + (3.5 + 6.06i)37-s + (−4.5 + 7.79i)41-s + (5.5 + 9.52i)43-s + (−6.5 + 2.59i)49-s + ⋯
L(s)  = 1  + 1.34·5-s + (0.188 + 0.981i)7-s − 0.904·11-s + (−0.693 + 1.20i)13-s + (0.363 − 0.630i)17-s + (0.573 + 0.993i)19-s − 0.625·23-s + 0.800·25-s + (−0.278 − 0.482i)29-s + (−0.359 − 0.622i)31-s + (0.253 + 1.31i)35-s + (0.575 + 0.996i)37-s + (−0.702 + 1.21i)41-s + (0.838 + 1.45i)43-s + (−0.928 + 0.371i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.296 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.296 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.296 - 0.954i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1873, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.296 - 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.785916447\)
\(L(\frac12)\) \(\approx\) \(1.785916447\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 2.59i)T \)
good5 \( 1 - 3T + 5T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + (2.5 - 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 3T + 23T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.5 - 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.5 - 9.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (5.5 - 9.52i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4 + 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.5 + 2.59i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.5 - 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.211469072734228254881094324105, −8.122894446217551394961610084594, −7.59759424138145145550802511522, −6.40065857029883971006959080547, −5.93744544711787277488291436519, −5.19870034944944628669699954424, −4.52546400025608236812179144421, −3.05401588831580605209632507910, −2.29445083035991655851967402962, −1.58430333699523206065321024565, 0.51093835662034441770303603523, 1.79509243681495243575394538969, 2.69108871259168185271216552364, 3.64222240930301489433243536970, 4.84644616104127240115935144689, 5.45475254456205444290706415813, 6.02872741454883456473477649065, 7.26722923731764579120596515264, 7.48607038340698147979430867781, 8.540529748498782536428073311394

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