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Properties

Label 2-504-21.2-c0-0-1
Degree $2$
Conductor $504$
Sign $0.475 + 0.879i$
Analytic cond. $0.251528$
Root an. cond. $0.501526$
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  + (−1.22 − 0.707i)5-s + (0.5 − 0.866i)7-s + (1.22 − 0.707i)11-s − 13-s + (0.5 − 0.866i)19-s + (0.499 + 0.866i)25-s + (0.5 + 0.866i)31-s + (−1.22 + 0.707i)35-s + (−0.5 + 0.866i)37-s + 1.41i·41-s − 43-s + (1.22 + 0.707i)47-s + (−0.499 − 0.866i)49-s − 2·55-s + (1.22 + 0.707i)65-s + ⋯
L(s)  = 1  + (−1.22 − 0.707i)5-s + (0.5 − 0.866i)7-s + (1.22 − 0.707i)11-s − 13-s + (0.5 − 0.866i)19-s + (0.499 + 0.866i)25-s + (0.5 + 0.866i)31-s + (−1.22 + 0.707i)35-s + (−0.5 + 0.866i)37-s + 1.41i·41-s − 43-s + (1.22 + 0.707i)47-s + (−0.499 − 0.866i)49-s − 2·55-s + (1.22 + 0.707i)65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.475 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.475 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.475 + 0.879i$
Analytic conductor: \(0.251528\)
Root analytic conductor: \(0.501526\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :0),\ 0.475 + 0.879i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7620268293\)
\(L(\frac12)\) \(\approx\) \(0.7620268293\)
\(L(1)\) 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 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + T + T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - 1.41iT - T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + 1.41iT - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - 1.41iT - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−11.28622041011387475511905024813, −10.11724510002133269095910640477, −9.051547773268775539995843567603, −8.267986284762705842757313145066, −7.44815481835568423864427563048, −6.60821740181229889229126340408, −4.98073153966592754997594122412, −4.33329557414190601727818274402, −3.27592709468854609113046103458, −1.08160545305569271639776234426, 2.13280923285315758235947114459, 3.54170474871652206340908868216, 4.47449960971579833236472073341, 5.70151990847289172330832249379, 6.97827839854047070715630623545, 7.55917077529776587537660342935, 8.549527855552450035468459675980, 9.499968631698451319253150087457, 10.45522636167720713992185071297, 11.66011663630116451726390585150

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