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Properties

Label 2-504-168.107-c1-0-21
Degree $2$
Conductor $504$
Sign $0.889 - 0.456i$
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  + (1.31 + 0.525i)2-s + (1.44 + 1.38i)4-s + (1.51 − 2.62i)5-s + (1.48 + 2.18i)7-s + (1.17 + 2.57i)8-s + (3.36 − 2.64i)10-s + (−4.18 + 2.41i)11-s − 1.60i·13-s + (0.806 + 3.65i)14-s + (0.190 + 3.99i)16-s + (5.79 − 3.34i)17-s + (−0.663 + 1.14i)19-s + (5.81 − 1.70i)20-s + (−6.75 + 0.971i)22-s + (4.32 − 7.49i)23-s + ⋯
L(s)  = 1  + (0.928 + 0.371i)2-s + (0.723 + 0.690i)4-s + (0.677 − 1.17i)5-s + (0.563 + 0.826i)7-s + (0.415 + 0.909i)8-s + (1.06 − 0.837i)10-s + (−1.26 + 0.727i)11-s − 0.445i·13-s + (0.215 + 0.976i)14-s + (0.0475 + 0.998i)16-s + (1.40 − 0.812i)17-s + (−0.152 + 0.263i)19-s + (1.30 − 0.381i)20-s + (−1.44 + 0.207i)22-s + (0.902 − 1.56i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.889 - 0.456i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.889 - 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.889 - 0.456i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.889 - 0.456i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.69726 + 0.651933i\)
\(L(\frac12)\) \(\approx\) \(2.69726 + 0.651933i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.31 - 0.525i)T \)
3 \( 1 \)
7 \( 1 + (-1.48 - 2.18i)T \)
good5 \( 1 + (-1.51 + 2.62i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.18 - 2.41i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.60iT - 13T^{2} \)
17 \( 1 + (-5.79 + 3.34i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.663 - 1.14i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.32 + 7.49i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.28T + 29T^{2} \)
31 \( 1 + (6.89 - 3.98i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.70 - 1.55i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 5.27iT - 41T^{2} \)
43 \( 1 + 3.12T + 43T^{2} \)
47 \( 1 + (-0.262 + 0.454i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.00 + 8.66i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.73 + 1.00i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.83 + 5.10i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.979 - 1.69i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.66T + 71T^{2} \)
73 \( 1 + (1.96 + 3.40i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.66 + 1.54i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.22iT - 83T^{2} \)
89 \( 1 + (9.40 + 5.42i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 15.3T + 97T^{2} \)
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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

−11.19039235110700880319166458440, −10.12378832902177503351712876481, −9.049039113698594727518608881270, −8.153343280714022009632176243871, −7.40660330032237345916366435219, −5.94970144642445772276760867355, −5.10107219204929057652474773335, −4.88800425340693238881710632800, −3.02971097804148984201338791421, −1.84838779248647126020262513533, 1.68380291637303176668483657462, 2.98913320364052736252447965283, 3.84919051757108791890304552099, 5.36963088942778717836350256274, 5.89406161348146816969230905000, 7.19799569201332264005717952751, 7.69671594914695020073949102316, 9.461524476119180959761102459938, 10.39161769095026849107646604695, 10.89492380780677144055725105770

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