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Properties

Label 2-504-63.38-c1-0-21
Degree $2$
Conductor $504$
Sign $-0.393 + 0.919i$
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 − 0.299i)3-s + (1.76 − 3.06i)5-s + (1.34 − 2.27i)7-s + (2.82 + 1.02i)9-s + (−1.57 + 0.909i)11-s + (−2.78 + 1.60i)13-s + (−3.93 + 4.69i)15-s + (2.93 − 5.08i)17-s + (−2.09 + 1.20i)19-s + (−2.97 + 3.48i)21-s + (−3.33 − 1.92i)23-s + (−3.74 − 6.48i)25-s + (−4.50 − 2.58i)27-s + (5.79 + 3.34i)29-s − 4.87i·31-s + ⋯
L(s)  = 1  + (−0.984 − 0.172i)3-s + (0.790 − 1.36i)5-s + (0.508 − 0.861i)7-s + (0.940 + 0.340i)9-s + (−0.474 + 0.274i)11-s + (−0.771 + 0.445i)13-s + (−1.01 + 1.21i)15-s + (0.711 − 1.23i)17-s + (−0.479 + 0.277i)19-s + (−0.649 + 0.760i)21-s + (−0.694 − 0.401i)23-s + (−0.748 − 1.29i)25-s + (−0.867 − 0.497i)27-s + (1.07 + 0.620i)29-s − 0.875i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.393 + 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.393 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.393 + 0.919i$
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: \((2,\ 504,\ (\ :1/2),\ -0.393 + 0.919i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.587606 - 0.891132i\)
\(L(\frac12)\) \(\approx\) \(0.587606 - 0.891132i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.70 + 0.299i)T \)
7 \( 1 + (-1.34 + 2.27i)T \)
good5 \( 1 + (-1.76 + 3.06i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.57 - 0.909i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.78 - 1.60i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.93 + 5.08i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.09 - 1.20i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.33 + 1.92i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.79 - 3.34i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.87iT - 31T^{2} \)
37 \( 1 + (0.905 + 1.56i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.03 + 8.71i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.36 - 4.09i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 7.92T + 47T^{2} \)
53 \( 1 + (2.26 + 1.30i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 8.81T + 59T^{2} \)
61 \( 1 - 10.3iT - 61T^{2} \)
67 \( 1 + 7.84T + 67T^{2} \)
71 \( 1 - 7.62iT - 71T^{2} \)
73 \( 1 + (11.7 + 6.79i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 1.84T + 79T^{2} \)
83 \( 1 + (-5.61 + 9.72i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.89 - 11.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-13.7 - 7.96i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.41136762100351389614118523858, −10.01636774390103025728811777114, −8.959680339160297235868293628427, −7.79653761336199149646324511863, −6.98910346734397915981170689487, −5.74113828793885852731911750389, −4.95722282765814429701479972729, −4.36231731766315161259551534264, −2.00517340462920481252802240296, −0.72623275887376279445049999063, 1.96374344515880747856257594615, 3.18674393789526767388365082993, 4.82070017136641259045891216268, 5.81056736979621516916154512584, 6.30011182636760045932188199053, 7.40675970424619663884330263022, 8.477539162188679802529133103927, 9.978194516064153250062158473074, 10.23888721736726788872665429703, 11.08046910769923323788552781334

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