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Properties

Label 2-504-7.4-c1-0-9
Degree $2$
Conductor $504$
Sign $-0.701 + 0.712i$
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  + (−0.5 − 0.866i)5-s + (−2 − 1.73i)7-s + (1.5 − 2.59i)11-s − 6·13-s + (−2.5 + 4.33i)17-s + (−0.5 − 0.866i)19-s + (−3.5 − 6.06i)23-s + (2 − 3.46i)25-s − 2·29-s + (2.5 − 4.33i)31-s + (−0.499 + 2.59i)35-s + (−1.5 − 2.59i)37-s + 2·41-s − 4·43-s + (2.5 + 4.33i)47-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (−0.755 − 0.654i)7-s + (0.452 − 0.783i)11-s − 1.66·13-s + (−0.606 + 1.05i)17-s + (−0.114 − 0.198i)19-s + (−0.729 − 1.26i)23-s + (0.400 − 0.692i)25-s − 0.371·29-s + (0.449 − 0.777i)31-s + (−0.0845 + 0.439i)35-s + (−0.246 − 0.427i)37-s + 0.312·41-s − 0.609·43-s + (0.364 + 0.631i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.701 + 0.712i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.701 + 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.266315 - 0.635470i\)
\(L(\frac12)\) \(\approx\) \(0.266315 - 0.635470i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (2 + 1.73i)T \)
good5 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 6T + 13T^{2} \)
17 \( 1 + (2.5 - 4.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.5 + 6.06i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + (-2.5 + 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (-2.5 - 4.33i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.5 + 12.9i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.5 + 7.79i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (3.5 - 6.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (-3.5 - 6.06i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−10.47411426907964889454806856143, −9.791607641016762117965858236127, −8.789028958898603649441060969282, −7.935896359213675516277942525339, −6.84338532292723554744468877781, −6.09122931317079614960196505957, −4.68001347949940271995915261146, −3.85090533475972350874847749862, −2.43408927175023092257506212239, −0.38567653674790494324807772002, 2.19344941765543884341902698282, 3.27339628949014245546500783653, 4.63796167885260613766022337859, 5.62970814769919942823310160566, 6.97354850770304209183758257008, 7.28969035610257481088852742156, 8.730102225013387808816326458930, 9.652635685990278623902713620372, 10.06016489954286374747617162633, 11.49549802431122192381217601787

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