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Properties

Label 2-504-63.38-c1-0-2
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: $\chi_{504} (353, \cdot )$
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

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

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