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Properties

Label 2-504-72.13-c1-0-69
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: $\chi_{504} (85, \cdot )$
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

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

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