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Properties

Label 2-504-63.5-c1-0-19
Degree $2$
Conductor $504$
Sign $-0.737 + 0.674i$
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  + (0.872 + 1.49i)3-s + (−2.09 − 3.63i)5-s + (−2.61 + 0.422i)7-s + (−1.47 + 2.61i)9-s + (1.44 + 0.834i)11-s + (−5.64 − 3.25i)13-s + (3.60 − 6.31i)15-s + (−1.45 − 2.52i)17-s + (−2.39 − 1.38i)19-s + (−2.91 − 3.53i)21-s + (−1.92 + 1.10i)23-s + (−6.31 + 10.9i)25-s + (−5.19 + 0.0719i)27-s + (5.69 − 3.28i)29-s − 0.478i·31-s + ⋯
L(s)  = 1  + (0.503 + 0.863i)3-s + (−0.938 − 1.62i)5-s + (−0.987 + 0.159i)7-s + (−0.491 + 0.870i)9-s + (0.435 + 0.251i)11-s + (−1.56 − 0.903i)13-s + (0.931 − 1.63i)15-s + (−0.353 − 0.611i)17-s + (−0.549 − 0.317i)19-s + (−0.635 − 0.772i)21-s + (−0.400 + 0.231i)23-s + (−1.26 + 2.18i)25-s + (−0.999 + 0.0138i)27-s + (1.05 − 0.610i)29-s − 0.0859i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.737 + 0.674i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.737 + 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.737 + 0.674i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.737 + 0.674i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.147336 - 0.379387i\)
\(L(\frac12)\) \(\approx\) \(0.147336 - 0.379387i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.872 - 1.49i)T \)
7 \( 1 + (2.61 - 0.422i)T \)
good5 \( 1 + (2.09 + 3.63i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.44 - 0.834i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (5.64 + 3.25i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.45 + 2.52i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.39 + 1.38i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.92 - 1.10i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.69 + 3.28i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.478iT - 31T^{2} \)
37 \( 1 + (0.378 - 0.655i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.769 + 1.33i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.79 + 8.31i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 8.11T + 47T^{2} \)
53 \( 1 + (7.11 - 4.10i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 0.853T + 59T^{2} \)
61 \( 1 - 4.50iT - 61T^{2} \)
67 \( 1 - 15.3T + 67T^{2} \)
71 \( 1 - 1.89iT - 71T^{2} \)
73 \( 1 + (6.22 - 3.59i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 11.0T + 79T^{2} \)
83 \( 1 + (0.162 + 0.280i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.86 + 4.95i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.22 - 2.44i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.27565259336136309024892079082, −9.565018373933457776197114812242, −8.901513789285127064900597294596, −8.118819490387680330590317319886, −7.17303354768480222016521496966, −5.50788222668914222560700037671, −4.69366924014929364766602227780, −3.92399898622520031450756312596, −2.62905650239720703966085443770, −0.21211978775576249955444283038, 2.33610544160963577764888922638, 3.22262192243594449211984283242, 4.17586841834775851291658784813, 6.37759165556104050155516546548, 6.68856225540551015202581008805, 7.45339898388350627632641242548, 8.361392530705279176537492640242, 9.540895491327002627333965813582, 10.37354568826241445195074421426, 11.38636516128546726312346243120

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