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Properties

Label 2-504-9.4-c1-0-12
Degree $2$
Conductor $504$
Sign $1$
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  + (1.11 + 1.32i)3-s + (1.43 − 2.49i)5-s + (−0.5 − 0.866i)7-s + (−0.520 + 2.95i)9-s + (0.592 + 1.02i)11-s + (2.37 − 4.12i)13-s + (4.91 − 0.866i)15-s + 5.41·17-s + 1.10·19-s + (0.592 − 1.62i)21-s + (−2.95 + 5.11i)23-s + (−1.64 − 2.84i)25-s + (−4.5 + 2.59i)27-s + (−2.49 − 4.31i)29-s + (2.78 − 4.82i)31-s + ⋯
L(s)  = 1  + (0.642 + 0.766i)3-s + (0.643 − 1.11i)5-s + (−0.188 − 0.327i)7-s + (−0.173 + 0.984i)9-s + (0.178 + 0.309i)11-s + (0.659 − 1.14i)13-s + (1.26 − 0.223i)15-s + 1.31·17-s + 0.253·19-s + (0.129 − 0.355i)21-s + (−0.615 + 1.06i)23-s + (−0.329 − 0.569i)25-s + (−0.866 + 0.499i)27-s + (−0.462 − 0.801i)29-s + (0.500 − 0.866i)31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.97481\)
\(L(\frac12)\) \(\approx\) \(1.97481\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.11 - 1.32i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (-1.43 + 2.49i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.592 - 1.02i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.37 + 4.12i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 5.41T + 17T^{2} \)
19 \( 1 - 1.10T + 19T^{2} \)
23 \( 1 + (2.95 - 5.11i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.49 + 4.31i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.78 + 4.82i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.42T + 37T^{2} \)
41 \( 1 + (3.81 - 6.61i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.78 - 6.55i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.141 + 0.245i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 4.22T + 53T^{2} \)
59 \( 1 + (5.86 - 10.1i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.992 + 1.71i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.76 + 11.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.11T + 71T^{2} \)
73 \( 1 + 0.327T + 73T^{2} \)
79 \( 1 + (-5.24 - 9.08i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.62 + 6.28i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 14.7T + 89T^{2} \)
97 \( 1 + (9.04 + 15.6i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.64801881568062147750559432836, −9.689966219888660801765952797025, −9.486653734529703568119013503777, −8.205430091053355367752803736005, −7.73867087884380480464311255400, −5.93428098205416709517535370126, −5.25426861693353173958739864670, −4.13781827320797941547118795192, −3.08313116286369443700034878597, −1.40121906865435043672717148394, 1.65958062063998843489816915992, 2.81590973226911545333304764240, 3.73938250427486782355641103056, 5.63908172909303715452789852848, 6.53400361979604955585751058407, 7.06227836801114093530091969354, 8.276925895457336461886466769818, 9.070469338324904231909199166086, 9.983341486684717662991122119401, 10.86678809587354435708560551334

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