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Properties

Label 1-3024-3024.1525-r1-0-0
Degree $1$
Conductor $3024$
Sign $0.328 + 0.944i$
Analytic cond. $324.973$
Root an. cond. $324.973$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.984 − 0.173i)5-s + (0.984 + 0.173i)11-s + (0.342 − 0.939i)13-s + (0.5 + 0.866i)17-s + (−0.866 − 0.5i)19-s + (−0.766 + 0.642i)23-s + (0.939 − 0.342i)25-s + (0.342 + 0.939i)29-s + (−0.766 + 0.642i)31-s + (0.866 − 0.5i)37-s + (−0.939 − 0.342i)41-s + (0.984 + 0.173i)43-s + (−0.766 − 0.642i)47-s + i·53-s + 55-s + ⋯
L(s)  = 1  + (0.984 − 0.173i)5-s + (0.984 + 0.173i)11-s + (0.342 − 0.939i)13-s + (0.5 + 0.866i)17-s + (−0.866 − 0.5i)19-s + (−0.766 + 0.642i)23-s + (0.939 − 0.342i)25-s + (0.342 + 0.939i)29-s + (−0.766 + 0.642i)31-s + (0.866 − 0.5i)37-s + (−0.939 − 0.342i)41-s + (0.984 + 0.173i)43-s + (−0.766 − 0.642i)47-s + i·53-s + 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.328 + 0.944i$
Analytic conductor: \(324.973\)
Root analytic conductor: \(324.973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1525, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (1:\ ),\ 0.328 + 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.167322532 + 1.541185001i\)
\(L(\frac12)\) \(\approx\) \(2.167322532 + 1.541185001i\)
\(L(1)\) \(\approx\) \(1.326565715 + 0.09918124984i\)
\(L(1)\) \(\approx\) \(1.326565715 + 0.09918124984i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.984 - 0.173i)T \)
11 \( 1 + (0.984 + 0.173i)T \)
13 \( 1 + (0.342 - 0.939i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (-0.766 + 0.642i)T \)
29 \( 1 + (0.342 + 0.939i)T \)
31 \( 1 + (-0.766 + 0.642i)T \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 + (-0.939 - 0.342i)T \)
43 \( 1 + (0.984 + 0.173i)T \)
47 \( 1 + (-0.766 - 0.642i)T \)
53 \( 1 + iT \)
59 \( 1 + (0.984 - 0.173i)T \)
61 \( 1 + (-0.642 + 0.766i)T \)
67 \( 1 + (-0.342 + 0.939i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (-0.939 + 0.342i)T \)
83 \( 1 + (-0.342 - 0.939i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.173 + 0.984i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.65044924742698985219458279210, −18.14481327611031966291442403873, −17.24068120315357629471423417110, −16.673181506337366448508736357210, −16.22589933342838084377532648688, −15.04163843885147156827610218351, −14.38161926960767218031674893698, −13.95374486618749749131845374449, −13.23648912309232666780708750972, −12.37248396935001601104241100788, −11.62451354135432333035049677147, −10.997922782984413228596734014968, −9.98386691416044630649296816290, −9.56733957225280430003639391188, −8.825953369518334238722916951073, −8.028387986169609055925975959260, −6.977872127222130410478644519644, −6.279454329998158670980012043219, −5.90333978514031332446220520080, −4.73511725334167322485494934027, −4.07218870978359003281762858218, −3.10124991270090608309569290801, −2.10664694195126617112707820320, −1.54093718582246080235448527435, −0.409673904610416408627117359518, 1.00573491132522660362403959612, 1.609367924554429919851330986804, 2.53956776437728416975382183257, 3.52223465703657972351942328407, 4.273105244192897074121520501028, 5.341049595175263685358206094962, 5.87805161933364564637768006070, 6.59091365293586441552620690180, 7.410160682696709191449838633580, 8.49850521840605964558430853527, 8.90284723313699801620499722987, 9.8417823306475358130809569564, 10.38920543458435896203026215722, 11.09646545363603167843482417308, 12.09331447752104547051179409147, 12.80263665145437204974663628396, 13.250069433949250162858804542229, 14.27478270917312534089696023294, 14.606868297996371919551869418781, 15.48992491358461357884309472987, 16.35927970513044209758322082735, 17.03434405468713949678157165991, 17.630453261926637854737543439178, 18.09048124417765108837996372265, 19.03051267995543220981472033153

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