Properties

Label 2-504-63.38-c1-0-10
Degree $2$
Conductor $504$
Sign $0.799 + 0.600i$
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  + (−1.54 − 0.791i)3-s + (−0.905 + 1.56i)5-s + (−2.60 − 0.449i)7-s + (1.74 + 2.43i)9-s + (−0.221 + 0.127i)11-s + (5.77 − 3.33i)13-s + (2.63 − 1.69i)15-s + (1.99 − 3.46i)17-s + (1.24 − 0.719i)19-s + (3.66 + 2.75i)21-s + (4.90 + 2.83i)23-s + (0.858 + 1.48i)25-s + (−0.758 − 5.14i)27-s + (−4.18 − 2.41i)29-s − 10.1i·31-s + ⋯
L(s)  = 1  + (−0.889 − 0.457i)3-s + (−0.405 + 0.701i)5-s + (−0.985 − 0.169i)7-s + (0.582 + 0.813i)9-s + (−0.0668 + 0.0385i)11-s + (1.60 − 0.925i)13-s + (0.681 − 0.438i)15-s + (0.484 − 0.839i)17-s + (0.286 − 0.165i)19-s + (0.798 + 0.601i)21-s + (1.02 + 0.590i)23-s + (0.171 + 0.297i)25-s + (−0.145 − 0.989i)27-s + (−0.777 − 0.448i)29-s − 1.82i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.799 + 0.600i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.799 + 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.799 + 0.600i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.799 + 0.600i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.862213 - 0.287791i\)
\(L(\frac12)\) \(\approx\) \(0.862213 - 0.287791i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.54 + 0.791i)T \)
7 \( 1 + (2.60 + 0.449i)T \)
good5 \( 1 + (0.905 - 1.56i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.221 - 0.127i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-5.77 + 3.33i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.99 + 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.24 + 0.719i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.90 - 2.83i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.18 + 2.41i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 10.1iT - 31T^{2} \)
37 \( 1 + (1.65 + 2.86i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.10 - 8.83i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.12 + 1.94i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 11.9T + 47T^{2} \)
53 \( 1 + (-3.97 - 2.29i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 5.11T + 59T^{2} \)
61 \( 1 + 9.93iT - 61T^{2} \)
67 \( 1 - 1.92T + 67T^{2} \)
71 \( 1 - 7.31iT - 71T^{2} \)
73 \( 1 + (2.47 + 1.43i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 3.66T + 79T^{2} \)
83 \( 1 + (2.68 - 4.64i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.378 - 0.655i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.21 + 2.43i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.12012106773007325520767754422, −10.13532389679203916598169046481, −9.192097532362624307695039079160, −7.73078829070985124714493055585, −7.20311614233446772864138808665, −6.15098531510278509986908551105, −5.52080162383295455369167196421, −3.92115803327291656950968835295, −2.89023872859941091669959248819, −0.799576619617029734621118091046, 1.11218039129143264941470643176, 3.44889503202385489193146746617, 4.25074495451993528419522812788, 5.45526221275477168326363728070, 6.27159379849875345222576607567, 7.13137884751142903078200154292, 8.776570445658878186445736520880, 9.013455505149345051182402185477, 10.36741037919242630562579633641, 10.86945441702614711185543669123

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