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Properties

Label 2-504-9.7-c1-0-11
Degree $2$
Conductor $504$
Sign $-0.500 + 0.866i$
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  + (−1.70 − 0.300i)3-s + (−0.266 − 0.460i)5-s + (−0.5 + 0.866i)7-s + (2.81 + 1.02i)9-s + (1.11 − 1.92i)11-s + (−1.03 − 1.78i)13-s + (0.315 + 0.866i)15-s + 0.815·17-s − 7.94·19-s + (1.11 − 1.32i)21-s + (−3.40 − 5.88i)23-s + (2.35 − 4.08i)25-s + (−4.49 − 2.59i)27-s + (3.73 − 6.47i)29-s + (−1.14 − 1.98i)31-s + ⋯
L(s)  = 1  + (−0.984 − 0.173i)3-s + (−0.118 − 0.206i)5-s + (−0.188 + 0.327i)7-s + (0.939 + 0.342i)9-s + (0.335 − 0.581i)11-s + (−0.286 − 0.495i)13-s + (0.0813 + 0.223i)15-s + 0.197·17-s − 1.82·19-s + (0.242 − 0.289i)21-s + (−0.709 − 1.22i)23-s + (0.471 − 0.816i)25-s + (−0.866 − 0.499i)27-s + (0.694 − 1.20i)29-s + (−0.205 − 0.356i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.500 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.500 + 0.866i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.500 + 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.301005 - 0.521356i\)
\(L(\frac12)\) \(\approx\) \(0.301005 - 0.521356i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.70 + 0.300i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (0.266 + 0.460i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.11 + 1.92i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.03 + 1.78i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 0.815T + 17T^{2} \)
19 \( 1 + 7.94T + 19T^{2} \)
23 \( 1 + (3.40 + 5.88i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.73 + 6.47i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.14 + 1.98i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 10.2T + 37T^{2} \)
41 \( 1 + (-1.29 - 2.24i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.145 - 0.251i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.213 - 0.368i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 1.41T + 53T^{2} \)
59 \( 1 + (1.71 + 2.97i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.23 + 9.07i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.10 - 3.64i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.0T + 71T^{2} \)
73 \( 1 + 9.09T + 73T^{2} \)
79 \( 1 + (3.73 - 6.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.76 + 15.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 2.09T + 89T^{2} \)
97 \( 1 + (5.94 - 10.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.60014510261294532481713840712, −10.04730335623560018130944252192, −8.706891578205801321247850695513, −7.999059219724100795637667591693, −6.59339765029298429041768913466, −6.14669608255171430775090568973, −4.97339891385591833001317338849, −4.01630718081834642589895139435, −2.28077332617000661074182925903, −0.40317102394608765792438332219, 1.69331867494190799890805405217, 3.63636952428341805729412293604, 4.57739172824763447881262398016, 5.61106369433337629090361547197, 6.77758558147399023190987384229, 7.18024020851259773208602792909, 8.644236699657362726540781940206, 9.653214072887572404314580949914, 10.47086525083394881033314956222, 11.09066192576382390608462877004

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