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Properties

Label 2-504-56.19-c1-0-26
Degree $2$
Conductor $504$
Sign $0.0481 + 0.998i$
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  + (−0.599 + 1.28i)2-s + (−1.28 − 1.53i)4-s + (1.03 − 1.78i)5-s + (−2.39 + 1.13i)7-s + (2.73 − 0.718i)8-s + (1.66 + 2.39i)10-s + (−0.982 − 1.70i)11-s − 4.20·13-s + (−0.0139 − 3.74i)14-s + (−0.720 + 3.93i)16-s + (3.09 − 1.78i)17-s + (−4.36 − 2.52i)19-s + (−4.06 + 0.703i)20-s + (2.76 − 0.237i)22-s + (−5.31 − 3.06i)23-s + ⋯
L(s)  = 1  + (−0.424 + 0.905i)2-s + (−0.640 − 0.768i)4-s + (0.461 − 0.799i)5-s + (−0.904 + 0.427i)7-s + (0.967 − 0.253i)8-s + (0.528 + 0.756i)10-s + (−0.296 − 0.512i)11-s − 1.16·13-s + (−0.00372 − 0.999i)14-s + (−0.180 + 0.983i)16-s + (0.751 − 0.434i)17-s + (−1.00 − 0.578i)19-s + (−0.909 + 0.157i)20-s + (0.590 − 0.0506i)22-s + (−1.10 − 0.639i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0481 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0481 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.0481 + 0.998i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.0481 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.372391 - 0.354868i\)
\(L(\frac12)\) \(\approx\) \(0.372391 - 0.354868i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.599 - 1.28i)T \)
3 \( 1 \)
7 \( 1 + (2.39 - 1.13i)T \)
good5 \( 1 + (-1.03 + 1.78i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.982 + 1.70i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.20T + 13T^{2} \)
17 \( 1 + (-3.09 + 1.78i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.36 + 2.52i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.31 + 3.06i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.34iT - 29T^{2} \)
31 \( 1 + (3.42 + 5.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.56 + 2.05i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.45iT - 41T^{2} \)
43 \( 1 - 4.33T + 43T^{2} \)
47 \( 1 + (4.88 - 8.46i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-11.2 + 6.48i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.17 + 3.56i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.03 + 1.78i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.77 - 4.80i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.44iT - 71T^{2} \)
73 \( 1 + (10.7 - 6.18i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.778 + 0.449i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.58iT - 83T^{2} \)
89 \( 1 + (12.6 + 7.30i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 0.550iT - 97T^{2} \)
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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

−10.19064722394699863426276835711, −9.690985334382532735718005774461, −8.912519538695451300035175354804, −8.080025813913210354926414401543, −7.04598336859188815664687254033, −5.99236567199626385947474534157, −5.37308576340842964223736477448, −4.23002751582933256201481852158, −2.37682280712872313238530768270, −0.33379205369016153081596012262, 1.94051070330680079743884024500, 3.04761376904751883147740677818, 4.03811967074687105474086875095, 5.45126217690318047559544082235, 6.80987571729800738236429179169, 7.49735388023252951471304608316, 8.669896330031259004988379809519, 9.806221961932217637733671289233, 10.22960517147352443700513231038, 10.73139509457293016063827042738

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