Properties

Degree $2$
Conductor $3024$
Sign $-0.968 + 0.248i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.465·5-s + (−0.656 − 2.56i)7-s + 5.28i·11-s − 4.50i·13-s − 3.15·17-s + 1.42i·19-s + 2.27i·23-s − 4.78·25-s − 6.09i·29-s − 2.76i·31-s + (−0.305 − 1.19i)35-s − 0.613·37-s + 6.93·41-s − 3.08·43-s − 9.30·47-s + ⋯
L(s)  = 1  + 0.208·5-s + (−0.248 − 0.968i)7-s + 1.59i·11-s − 1.24i·13-s − 0.766·17-s + 0.327i·19-s + 0.474i·23-s − 0.956·25-s − 1.13i·29-s − 0.497i·31-s + (−0.0516 − 0.201i)35-s − 0.100·37-s + 1.08·41-s − 0.470·43-s − 1.35·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.248i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.968 + 0.248i$
Motivic weight: \(1\)
Character: $\chi_{3024} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.968 + 0.248i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4173087410\)
\(L(\frac12)\) \(\approx\) \(0.4173087410\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.656 + 2.56i)T \)
good5 \( 1 - 0.465T + 5T^{2} \)
11 \( 1 - 5.28iT - 11T^{2} \)
13 \( 1 + 4.50iT - 13T^{2} \)
17 \( 1 + 3.15T + 17T^{2} \)
19 \( 1 - 1.42iT - 19T^{2} \)
23 \( 1 - 2.27iT - 23T^{2} \)
29 \( 1 + 6.09iT - 29T^{2} \)
31 \( 1 + 2.76iT - 31T^{2} \)
37 \( 1 + 0.613T + 37T^{2} \)
41 \( 1 - 6.93T + 41T^{2} \)
43 \( 1 + 3.08T + 43T^{2} \)
47 \( 1 + 9.30T + 47T^{2} \)
53 \( 1 + 12.1iT - 53T^{2} \)
59 \( 1 + 3.62T + 59T^{2} \)
61 \( 1 + 2.27iT - 61T^{2} \)
67 \( 1 + 13.3T + 67T^{2} \)
71 \( 1 - 7.38iT - 71T^{2} \)
73 \( 1 - 10.8iT - 73T^{2} \)
79 \( 1 + 1.17T + 79T^{2} \)
83 \( 1 - 15.0T + 83T^{2} \)
89 \( 1 + 5.39T + 89T^{2} \)
97 \( 1 + 10.4iT - 97T^{2} \)
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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

−8.101421400015040727709682828215, −7.69558911042321331940725274580, −6.92713133807997496609948175115, −6.18057277317816290155532248877, −5.25053414478620037498723168999, −4.38928182708376996434659285694, −3.73034717418982390415872884111, −2.56333204999907865273434210620, −1.57822780939865174197601192919, −0.12325388572855217142061574842, 1.54612959265250152506306548993, 2.59020155908153160690514841853, 3.38800125123405024970446940161, 4.44299111728369157422807346497, 5.31232221865920423751317669077, 6.21375447588889438865706969407, 6.49505564317947415951864285651, 7.62484432590216459675089232643, 8.582331775965785387493205600514, 8.999341542033722386502414223466

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