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Properties

Label 2-504-56.53-c1-0-7
Degree $2$
Conductor $504$
Sign $0.465 - 0.885i$
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  + (−1.38 − 0.276i)2-s + (1.84 + 0.767i)4-s + (0.476 − 0.274i)5-s + (−2.60 + 0.447i)7-s + (−2.34 − 1.57i)8-s + (−0.736 + 0.249i)10-s + (2.07 + 1.19i)11-s + 3.96i·13-s + (3.74 + 0.100i)14-s + (2.82 + 2.83i)16-s + (−2.10 + 3.65i)17-s + (5.75 − 3.32i)19-s + (1.09 − 0.142i)20-s + (−2.54 − 2.23i)22-s + (−1.17 − 2.03i)23-s + ⋯
L(s)  = 1  + (−0.980 − 0.195i)2-s + (0.923 + 0.383i)4-s + (0.212 − 0.122i)5-s + (−0.985 + 0.169i)7-s + (−0.830 − 0.557i)8-s + (−0.232 + 0.0788i)10-s + (0.624 + 0.360i)11-s + 1.10i·13-s + (0.999 + 0.0268i)14-s + (0.705 + 0.708i)16-s + (−0.511 + 0.885i)17-s + (1.31 − 0.761i)19-s + (0.243 − 0.0317i)20-s + (−0.541 − 0.475i)22-s + (−0.244 − 0.424i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.465 - 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.465 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.465 - 0.885i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.465 - 0.885i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.654868 + 0.395537i\)
\(L(\frac12)\) \(\approx\) \(0.654868 + 0.395537i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 + 0.276i)T \)
3 \( 1 \)
7 \( 1 + (2.60 - 0.447i)T \)
good5 \( 1 + (-0.476 + 0.274i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.07 - 1.19i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.96iT - 13T^{2} \)
17 \( 1 + (2.10 - 3.65i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.75 + 3.32i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.17 + 2.03i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.21iT - 29T^{2} \)
31 \( 1 + (0.433 - 0.750i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.229 - 0.132i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.24T + 41T^{2} \)
43 \( 1 - 5.35iT - 43T^{2} \)
47 \( 1 + (-1.29 - 2.25i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-9.36 - 5.40i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.26 - 1.88i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.18 - 3.56i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.31 - 1.33i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.76T + 71T^{2} \)
73 \( 1 + (2.33 - 4.03i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.308 + 0.533i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.09iT - 83T^{2} \)
89 \( 1 + (3.19 + 5.53i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 12.9T + 97T^{2} \)
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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

−10.97441765854563840194225307443, −9.976370915299730921924966828549, −9.209870795716432553580238855660, −8.835050635268126192592786307222, −7.34994018275957213648898286830, −6.75191382041786705565826616871, −5.79554758052802207113566936110, −4.10410819365490060349487932762, −2.88371558314361402573664532330, −1.49210047783269500508069196627, 0.65291035179945985087707469087, 2.51327830891572903416252273874, 3.65217276353928316764739625196, 5.56213512890928647362483783252, 6.24118820591826091320572571394, 7.26153937734271502713115846309, 8.038138743798539979273492844625, 9.180133472838566418715021112266, 9.810574213345601210516927763818, 10.42897663212600328211317456954

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