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Properties

Label 2-504-7.2-c1-0-5
Degree $2$
Conductor $504$
Sign $0.858 + 0.513i$
Analytic cond. $4.02446$
Root an. cond. $2.00610$
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  + (1.63 − 2.83i)5-s + (−1.5 + 2.17i)7-s + (1.63 + 2.83i)11-s + 6.27·13-s + (−2 − 3.46i)17-s + (3.13 − 5.43i)19-s + (2 − 3.46i)23-s + (−2.86 − 4.95i)25-s − 5.27·29-s + (0.5 + 0.866i)31-s + (3.72 + 7.82i)35-s + (1.13 − 1.97i)37-s + 4.54·41-s + 0.274·43-s + (−3 + 5.19i)47-s + ⋯
L(s)  = 1  + (0.732 − 1.26i)5-s + (−0.566 + 0.823i)7-s + (0.493 + 0.855i)11-s + 1.74·13-s + (−0.485 − 0.840i)17-s + (0.719 − 1.24i)19-s + (0.417 − 0.722i)23-s + (−0.572 − 0.991i)25-s − 0.979·29-s + (0.0898 + 0.155i)31-s + (0.629 + 1.32i)35-s + (0.186 − 0.323i)37-s + 0.710·41-s + 0.0419·43-s + (−0.437 + 0.757i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.858 + 0.513i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.858 + 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.858 + 0.513i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.858 + 0.513i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57944 - 0.436629i\)
\(L(\frac12)\) \(\approx\) \(1.57944 - 0.436629i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 - 2.17i)T \)
good5 \( 1 + (-1.63 + 2.83i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.63 - 2.83i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 6.27T + 13T^{2} \)
17 \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.13 + 5.43i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.27T + 29T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.13 + 1.97i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.54T + 41T^{2} \)
43 \( 1 - 0.274T + 43T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.63 - 8.03i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.637 + 1.10i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.137 - 0.238i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + (-2.13 - 3.70i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.77 - 10.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.27T + 83T^{2} \)
89 \( 1 + (5.27 - 9.13i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 8.72T + 97T^{2} \)
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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

−10.91891557843543354682640306449, −9.493548666382581134126289624919, −9.203677203850955063982623019990, −8.537361111344313426738922788366, −7.06631439851383122304090628705, −6.07745464313649604759445013024, −5.24944313869593765526938279797, −4.24061181184462077621066371421, −2.64935304768218923267406787111, −1.21089466871120919642379493177, 1.52825652717293792338932886496, 3.32204951488817769804452977735, 3.77862539058331446078177189146, 5.87346250481402089294656252869, 6.23621073976120312592508574383, 7.18527564783267972656927301905, 8.299087627165461887866300985473, 9.388035396918880554245982036838, 10.25229424608961117732519219183, 10.91438935129028814153222311774

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