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Properties

Label 2-504-56.53-c1-0-19
Degree $2$
Conductor $504$
Sign $0.967 + 0.254i$
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  + (−1.36 + 0.378i)2-s + (1.71 − 1.03i)4-s + (0.586 − 0.338i)5-s + (2.23 − 1.41i)7-s + (−1.94 + 2.05i)8-s + (−0.670 + 0.683i)10-s + (1.44 + 0.835i)11-s + 1.28i·13-s + (−2.51 + 2.77i)14-s + (1.86 − 3.53i)16-s + (3.18 − 5.51i)17-s + (−2.20 + 1.27i)19-s + (0.655 − 1.18i)20-s + (−2.28 − 0.590i)22-s + (−0.127 − 0.221i)23-s + ⋯
L(s)  = 1  + (−0.963 + 0.267i)2-s + (0.856 − 0.516i)4-s + (0.262 − 0.151i)5-s + (0.845 − 0.534i)7-s + (−0.687 + 0.726i)8-s + (−0.212 + 0.216i)10-s + (0.436 + 0.251i)11-s + 0.355i·13-s + (−0.671 + 0.740i)14-s + (0.467 − 0.884i)16-s + (0.771 − 1.33i)17-s + (−0.506 + 0.292i)19-s + (0.146 − 0.265i)20-s + (−0.487 − 0.125i)22-s + (−0.0266 − 0.0461i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.254i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 + 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.967 + 0.254i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.967 + 0.254i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08816 - 0.140850i\)
\(L(\frac12)\) \(\approx\) \(1.08816 - 0.140850i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 - 0.378i)T \)
3 \( 1 \)
7 \( 1 + (-2.23 + 1.41i)T \)
good5 \( 1 + (-0.586 + 0.338i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.44 - 0.835i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.28iT - 13T^{2} \)
17 \( 1 + (-3.18 + 5.51i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.20 - 1.27i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.127 + 0.221i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.27iT - 29T^{2} \)
31 \( 1 + (-2.14 + 3.71i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.62 + 3.24i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.43T + 41T^{2} \)
43 \( 1 - 5.48iT - 43T^{2} \)
47 \( 1 + (-4.73 - 8.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.91 - 2.83i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (8.74 + 5.04i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-13.2 + 7.62i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (13.6 + 7.88i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.48T + 71T^{2} \)
73 \( 1 + (1.43 - 2.48i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.09 - 10.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.63iT - 83T^{2} \)
89 \( 1 + (-3.40 - 5.89i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 0.477T + 97T^{2} \)
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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

−10.87660581280346697302232211445, −9.657737795809455919902071434409, −9.361884018173718390152988444803, −8.000084674682883657038201925000, −7.56083166023515122729041795685, −6.45142607225395251422459391266, −5.43019606331641595475294737183, −4.22233973383906956713500220624, −2.41072205632550097996605099668, −1.07919194503014320321241634534, 1.38621553483583209119478960441, 2.60567605734904859148651252756, 3.97766679349258350079471452617, 5.56901540284606402195475274091, 6.44405679893766980341455372654, 7.61242861450308635665019567063, 8.454228358160756497558840151127, 9.003820451096654107542186529698, 10.27604196096036638072194549891, 10.66417350132914044839588609608

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