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Properties

Label 2-504-9.4-c1-0-5
Degree $2$
Conductor $504$
Sign $0.652 - 0.757i$
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  + (0.742 − 1.56i)3-s + (−1.76 + 3.05i)5-s + (0.5 + 0.866i)7-s + (−1.89 − 2.32i)9-s + (2.08 + 3.60i)11-s + (−1.52 + 2.63i)13-s + (3.46 + 5.02i)15-s + 3.79·17-s + 5.40·19-s + (1.72 − 0.139i)21-s + (−4.48 + 7.76i)23-s + (−3.71 − 6.42i)25-s + (−5.04 + 1.24i)27-s + (1.46 + 2.53i)29-s + (3.08 − 5.34i)31-s + ⋯
L(s)  = 1  + (0.428 − 0.903i)3-s + (−0.788 + 1.36i)5-s + (0.188 + 0.327i)7-s + (−0.632 − 0.774i)9-s + (0.628 + 1.08i)11-s + (−0.421 + 0.730i)13-s + (0.895 + 1.29i)15-s + 0.920·17-s + 1.24·19-s + (0.376 − 0.0304i)21-s + (−0.935 + 1.61i)23-s + (−0.742 − 1.28i)25-s + (−0.970 + 0.239i)27-s + (0.271 + 0.469i)29-s + (0.554 − 0.960i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.652 - 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.652 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.652 - 0.757i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.652 - 0.757i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25138 + 0.573464i\)
\(L(\frac12)\) \(\approx\) \(1.25138 + 0.573464i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.742 + 1.56i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (1.76 - 3.05i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.08 - 3.60i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.52 - 2.63i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.79T + 17T^{2} \)
19 \( 1 - 5.40T + 19T^{2} \)
23 \( 1 + (4.48 - 7.76i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.46 - 2.53i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.08 + 5.34i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.66T + 37T^{2} \)
41 \( 1 + (-1.83 + 3.18i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.31 - 2.28i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.860 + 1.49i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 11.4T + 53T^{2} \)
59 \( 1 + (-0.564 + 0.977i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.03 + 10.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.12 - 1.95i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.92T + 71T^{2} \)
73 \( 1 - 2.50T + 73T^{2} \)
79 \( 1 + (-3.16 - 5.48i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.52 - 9.56i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 13.4T + 89T^{2} \)
97 \( 1 + (9.46 + 16.4i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.46506099531947208823092457874, −10.01448588382802568860509128705, −9.343440830323410820711212214648, −7.982292906331459192469985546364, −7.34873718917302688454540023499, −6.84979174975172063626966740687, −5.65443654302340197996643326159, −3.97308205928744715123506891145, −3.02678212570781773191919939046, −1.78727963911706166774190667790, 0.840223686457268427511014195564, 3.09840044447219313625134524573, 4.07138090797011099467270691880, 4.91757802968895471076498627791, 5.80940352920155365044616885942, 7.56851424343093065518527046253, 8.348526454565621920168204478424, 8.818682275468812122043154933592, 9.900789560759875867852890709895, 10.64699203820654809868694864274

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