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Properties

Label 2-3024-9.4-c1-0-9
Degree $2$
Conductor $3024$
Sign $-0.939 - 0.342i$
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  + (−1.5 + 2.59i)5-s + (0.5 + 0.866i)7-s + (3 + 5.19i)11-s + (−1 + 1.73i)13-s − 6·17-s + 7·19-s + (−1.5 + 2.59i)23-s + (−2 − 3.46i)25-s + (3 + 5.19i)29-s + (1 − 1.73i)31-s − 3·35-s + 2·37-s + (1 + 1.73i)43-s + (−0.499 + 0.866i)49-s − 6·53-s + ⋯
L(s)  = 1  + (−0.670 + 1.16i)5-s + (0.188 + 0.327i)7-s + (0.904 + 1.56i)11-s + (−0.277 + 0.480i)13-s − 1.45·17-s + 1.60·19-s + (−0.312 + 0.541i)23-s + (−0.400 − 0.692i)25-s + (0.557 + 0.964i)29-s + (0.179 − 0.311i)31-s − 0.507·35-s + 0.328·37-s + (0.152 + 0.264i)43-s + (−0.0714 + 0.123i)49-s − 0.824·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.939 - 0.342i$
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: \((2,\ 3024,\ (\ :1/2),\ -0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.300030440\)
\(L(\frac12)\) \(\approx\) \(1.300030440\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 - 7T + 19T^{2} \)
23 \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1 - 1.73i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.5 + 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4 + 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + (-2.5 - 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.383126808480410077594299163292, −8.157009355248475792742284741318, −7.34642532321144877470488027067, −6.93494401502035205732516124128, −6.31459767718481154988040146181, −5.04125930076326715285816824818, −4.35764082976148223710878118128, −3.50402980389435609133345168730, −2.55299713738741700113025647450, −1.60510241191156399030294964624, 0.45477622755877449628922801115, 1.20092829442801032971968923783, 2.75735760847672058934669876033, 3.77975240139713523891249568697, 4.42231503339086593042553156349, 5.22078132431895077307511529352, 6.07175033094876052436970452198, 6.89384466156643867158957973411, 7.900083646337864072267200587299, 8.374203506294934692410051619554

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