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Properties

Label 2-504-63.5-c1-0-7
Degree $2$
Conductor $504$
Sign $0.982 - 0.188i$
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  + (0.589 − 1.62i)3-s + (1.11 + 1.92i)5-s + (−0.553 + 2.58i)7-s + (−2.30 − 1.92i)9-s + (1.83 + 1.06i)11-s + (5.10 + 2.94i)13-s + (3.79 − 0.676i)15-s + (2.34 + 4.05i)17-s + (−4.54 − 2.62i)19-s + (3.88 + 2.42i)21-s + (3.77 − 2.18i)23-s + (0.0173 − 0.0300i)25-s + (−4.48 + 2.62i)27-s + (−2.25 + 1.30i)29-s − 7.61i·31-s + ⋯
L(s)  = 1  + (0.340 − 0.940i)3-s + (0.498 + 0.863i)5-s + (−0.209 + 0.977i)7-s + (−0.768 − 0.640i)9-s + (0.553 + 0.319i)11-s + (1.41 + 0.817i)13-s + (0.981 − 0.174i)15-s + (0.567 + 0.983i)17-s + (−1.04 − 0.602i)19-s + (0.848 + 0.529i)21-s + (0.788 − 0.454i)23-s + (0.00346 − 0.00600i)25-s + (−0.863 + 0.504i)27-s + (−0.418 + 0.241i)29-s − 1.36i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.982 - 0.188i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.982 - 0.188i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73950 + 0.165367i\)
\(L(\frac12)\) \(\approx\) \(1.73950 + 0.165367i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.589 + 1.62i)T \)
7 \( 1 + (0.553 - 2.58i)T \)
good5 \( 1 + (-1.11 - 1.92i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.83 - 1.06i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-5.10 - 2.94i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.34 - 4.05i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.54 + 2.62i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.77 + 2.18i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.25 - 1.30i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.61iT - 31T^{2} \)
37 \( 1 + (1.80 - 3.12i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.0395 - 0.0684i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.24 + 2.16i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.79T + 47T^{2} \)
53 \( 1 + (-4.08 + 2.35i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 13.1T + 59T^{2} \)
61 \( 1 + 8.15iT - 61T^{2} \)
67 \( 1 + 4.75T + 67T^{2} \)
71 \( 1 - 10.0iT - 71T^{2} \)
73 \( 1 + (12.6 - 7.30i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 14.5T + 79T^{2} \)
83 \( 1 + (6.41 + 11.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.73 - 4.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-12.9 + 7.46i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.13824314410856716399703853240, −10.02097062630735407123603999373, −8.852419056963383472700734325852, −8.516433955110713748424237813888, −7.05158218445349248096381444969, −6.39342434928018127918947179702, −5.81090335057587641474287721434, −3.92357763237010267239638269103, −2.68819402651533776284678992553, −1.70844001131577607354080000459, 1.18314537972317454042422285505, 3.22243929564487657640376190986, 4.04684059474092325884538048978, 5.16826303337787520744351062232, 6.01139432937723986043391249142, 7.38593586227498047814329702276, 8.587836460093279751321059755079, 9.006033175959636443191162364407, 10.07275174585847300004809386504, 10.65914432675119107701750025692

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