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Properties

Label 2-504-7.4-c1-0-1
Degree $2$
Conductor $504$
Sign $-0.266 - 0.963i$
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 + (−2.5 + 0.866i)7-s + (−2.5 + 4.33i)11-s + 2·13-s + (−3 + 5.19i)17-s + (−1 − 1.73i)19-s + (3 + 5.19i)23-s + (2 − 3.46i)25-s − 3·29-s + (−2.5 + 4.33i)31-s + (−2 − 1.73i)35-s + (1 + 1.73i)37-s + 8·41-s − 4·43-s + (−2 − 3.46i)47-s + ⋯
L(s)  = 1  + (0.223 + 0.387i)5-s + (−0.944 + 0.327i)7-s + (−0.753 + 1.30i)11-s + 0.554·13-s + (−0.727 + 1.26i)17-s + (−0.229 − 0.397i)19-s + (0.625 + 1.08i)23-s + (0.400 − 0.692i)25-s − 0.557·29-s + (−0.449 + 0.777i)31-s + (−0.338 − 0.292i)35-s + (0.164 + 0.284i)37-s + 1.24·41-s − 0.609·43-s + (−0.291 − 0.505i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.266 - 0.963i$
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: \((2,\ 504,\ (\ :1/2),\ -0.266 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.607340 + 0.798338i\)
\(L(\frac12)\) \(\approx\) \(0.607340 + 0.798338i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 - 0.866i)T \)
good5 \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.5 - 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (2 + 3.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.5 - 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6 - 10.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (-7 + 12.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 17T + 83T^{2} \)
89 \( 1 + (9 + 15.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 3T + 97T^{2} \)
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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

−10.95634341454770104950836846532, −10.33153665707918213721554771598, −9.463200845042747180078195950763, −8.627389753104547939759017785006, −7.40941563614987038053699705727, −6.60662639000712338966927795390, −5.71777337773346049355343596544, −4.46963355997569490413701889847, −3.20927906090790748271329320923, −2.02216694401315799934367373563, 0.57935378938311712888551342100, 2.64449483988025639704890954775, 3.70610374434259922435854071636, 5.05621067840479532079886702507, 6.02071897137359704024113521651, 6.90028831221768740871836491647, 8.047871588492018681910055240308, 8.983874708821005765724252391969, 9.639249753096445109050294396276, 10.83938473008914582192044393326

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