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Properties

Label 2-504-63.47-c1-0-22
Degree $2$
Conductor $504$
Sign $-0.553 + 0.833i$
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  + (0.994 − 1.41i)3-s − 1.58·5-s + (−1.06 − 2.42i)7-s + (−1.02 − 2.81i)9-s + 3.95i·11-s + (5.09 − 2.93i)13-s + (−1.57 + 2.24i)15-s + (−3.19 − 5.53i)17-s + (−6.37 − 3.68i)19-s + (−4.49 − 0.898i)21-s − 2.43i·23-s − 2.48·25-s + (−5.01 − 1.35i)27-s + (4.34 + 2.50i)29-s + (0.855 + 0.493i)31-s + ⋯
L(s)  = 1  + (0.573 − 0.818i)3-s − 0.709·5-s + (−0.402 − 0.915i)7-s + (−0.341 − 0.939i)9-s + 1.19i·11-s + (1.41 − 0.815i)13-s + (−0.407 + 0.580i)15-s + (−0.775 − 1.34i)17-s + (−1.46 − 0.844i)19-s + (−0.980 − 0.196i)21-s − 0.506i·23-s − 0.496·25-s + (−0.965 − 0.260i)27-s + (0.806 + 0.465i)29-s + (0.153 + 0.0886i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 + 0.833i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.553 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.553 + 0.833i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (425, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.553 + 0.833i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.579483 - 1.08054i\)
\(L(\frac12)\) \(\approx\) \(0.579483 - 1.08054i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.994 + 1.41i)T \)
7 \( 1 + (1.06 + 2.42i)T \)
good5 \( 1 + 1.58T + 5T^{2} \)
11 \( 1 - 3.95iT - 11T^{2} \)
13 \( 1 + (-5.09 + 2.93i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.19 + 5.53i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.37 + 3.68i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 2.43iT - 23T^{2} \)
29 \( 1 + (-4.34 - 2.50i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.855 - 0.493i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.183 + 0.317i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.58 - 9.67i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.26 + 2.19i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.543 + 0.940i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.89 + 3.97i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.73 - 4.73i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-12.4 + 7.20i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.70 + 9.88i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.65iT - 71T^{2} \)
73 \( 1 + (-7.64 + 4.41i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.30 - 9.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.96 + 5.14i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.70 + 8.15i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.4 - 6.05i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.72423352564227356747159869362, −9.632929717188146495103779518218, −8.632932699632671263793076865746, −7.87978518281335697597665897036, −6.95645950884207819951871768142, −6.44391018907301030585710756706, −4.63324611236733027490267120552, −3.69251508990650808121981010935, −2.45419123989565681402532596920, −0.68266143797408834286875769168, 2.21090667879661392646653732764, 3.70706442244145766214785888814, 4.06603938816408116396671194905, 5.75416929746667024165961387256, 6.38112559371178912473313009938, 8.180342732140836004955494530293, 8.506474979866535941448289526937, 9.195929590600638704262330490344, 10.48474950254521797898194628904, 11.07029287174429795412935729309

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