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Properties

Label 2-504-56.37-c1-0-9
Degree $2$
Conductor $504$
Sign $0.861 + 0.508i$
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  + (−0.535 − 1.30i)2-s + (−1.42 + 1.40i)4-s + (−1.93 − 1.11i)5-s + (−0.5 + 2.59i)7-s + (2.59 + 1.11i)8-s + (−0.427 + 3.13i)10-s + (1.93 − 1.11i)11-s + (3.66 − 0.736i)14-s + (0.0729 − 3.99i)16-s + (1.73 + 3i)17-s + (6.70 + 3.87i)19-s + (4.33 − 1.11i)20-s + (−2.5 − 1.93i)22-s + (3.46 − 6i)23-s + (−2.92 − 4.40i)28-s + 2.23i·29-s + ⋯
L(s)  = 1  + (−0.378 − 0.925i)2-s + (−0.713 + 0.700i)4-s + (−0.866 − 0.499i)5-s + (−0.188 + 0.981i)7-s + (0.918 + 0.395i)8-s + (−0.135 + 0.990i)10-s + (0.583 − 0.337i)11-s + (0.980 − 0.196i)14-s + (0.0182 − 0.999i)16-s + (0.420 + 0.727i)17-s + (1.53 + 0.888i)19-s + (0.968 − 0.249i)20-s + (−0.533 − 0.412i)22-s + (0.722 − 1.25i)23-s + (−0.553 − 0.833i)28-s + 0.415i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 + 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.861 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.861 + 0.508i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.861 + 0.508i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.919235 - 0.251008i\)
\(L(\frac12)\) \(\approx\) \(0.919235 - 0.251008i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.535 + 1.30i)T \)
3 \( 1 \)
7 \( 1 + (0.5 - 2.59i)T \)
good5 \( 1 + (1.93 + 1.11i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.93 + 1.11i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + (-1.73 - 3i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.70 - 3.87i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.46 + 6i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.23iT - 29T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.70 - 3.87i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 10.3T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + (1.73 - 3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (9.68 - 5.59i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.93 + 1.11i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.70 - 3.87i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.70 + 3.87i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 + (-5 - 8.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.5 + 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 11.1iT - 83T^{2} \)
89 \( 1 + (6.92 - 12i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + T + 97T^{2} \)
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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

−11.02318389468497229535494925468, −9.888609509982890140732495887142, −9.105233433940507687400332115646, −8.344430479337587471424851803297, −7.66222341911397303714812069541, −6.11814265945284479968454210514, −4.89767286388241139039081893135, −3.82392804144644238671811658166, −2.82092461034985427421944540261, −1.12965146234047735711814096702, 0.877941589925355588623482634574, 3.35256571302802823548628651644, 4.35265963019640119225915143568, 5.46260436227634966840448415583, 6.81593480233986792169031447544, 7.35882326520100336473065977923, 7.894762801487939336625358360542, 9.430787580000023680403832827787, 9.684278348224833984841533655823, 11.08707918754753521442498279826

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