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Properties

Label 2-504-72.61-c1-0-12
Degree $2$
Conductor $504$
Sign $-0.956 - 0.291i$
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  + (1.37 + 0.329i)2-s + (−0.821 + 1.52i)3-s + (1.78 + 0.905i)4-s + (−2.72 + 1.57i)5-s + (−1.63 + 1.82i)6-s + (−0.5 + 0.866i)7-s + (2.15 + 1.83i)8-s + (−1.65 − 2.50i)9-s + (−4.25 + 1.26i)10-s + (−2.21 − 1.27i)11-s + (−2.84 + 1.97i)12-s + (−0.964 + 0.556i)13-s + (−0.972 + 1.02i)14-s + (−0.161 − 5.43i)15-s + (2.36 + 3.22i)16-s + 1.14·17-s + ⋯
L(s)  = 1  + (0.972 + 0.232i)2-s + (−0.474 + 0.880i)3-s + (0.891 + 0.452i)4-s + (−1.21 + 0.702i)5-s + (−0.665 + 0.746i)6-s + (−0.188 + 0.327i)7-s + (0.761 + 0.647i)8-s + (−0.550 − 0.834i)9-s + (−1.34 + 0.400i)10-s + (−0.667 − 0.385i)11-s + (−0.821 + 0.570i)12-s + (−0.267 + 0.154i)13-s + (−0.259 + 0.274i)14-s + (−0.0417 − 1.40i)15-s + (0.590 + 0.807i)16-s + 0.278·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 - 0.291i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.956 - 0.291i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.956 - 0.291i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.956 - 0.291i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.201100 + 1.34848i\)
\(L(\frac12)\) \(\approx\) \(0.201100 + 1.34848i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.37 - 0.329i)T \)
3 \( 1 + (0.821 - 1.52i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (2.72 - 1.57i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.21 + 1.27i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.964 - 0.556i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.14T + 17T^{2} \)
19 \( 1 - 4.46iT - 19T^{2} \)
23 \( 1 + (0.842 + 1.45i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.17 + 2.41i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.39 - 5.87i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 3.26iT - 37T^{2} \)
41 \( 1 + (1.46 + 2.53i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-9.14 - 5.28i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.80 - 10.0i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 9.23iT - 53T^{2} \)
59 \( 1 + (-11.4 + 6.63i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.64 + 1.52i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.47 + 5.46i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 + 7.16T + 73T^{2} \)
79 \( 1 + (2.48 - 4.29i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (10.8 + 6.29i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 6.04T + 89T^{2} \)
97 \( 1 + (-5.01 + 8.67i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.32922600592474579456714759458, −10.78784921726366635512364945437, −9.850608919754241227840022983140, −8.367010456260007788120232580201, −7.60780389836416940472973903969, −6.49311928393338876367562703735, −5.64027391106121715403973746485, −4.58034031084535069565538480197, −3.66294230717136913271301651281, −2.87401695897620878990087951047, 0.61254768525471118752209774243, 2.34729168795539468786353292601, 3.78197374009949214195691700887, 4.83857561170764493916541502559, 5.60250135101249151542311425458, 6.95602829499551806786320185369, 7.48482063253888597901869311725, 8.371222160159444140654413009535, 9.910010383470035588037294255519, 11.01139827524528449525271590228

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