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Properties

Label 2-504-63.20-c1-0-12
Degree $2$
Conductor $504$
Sign $0.0850 - 0.996i$
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  + (0.610 + 1.62i)3-s + (1.91 + 3.32i)5-s + (2.41 − 1.06i)7-s + (−2.25 + 1.97i)9-s + (0.585 + 0.338i)11-s + (4.22 − 2.43i)13-s + (−4.21 + 5.13i)15-s − 5.79·17-s − 4.22i·19-s + (3.21 + 3.26i)21-s + (4.76 − 2.75i)23-s + (−4.86 + 8.41i)25-s + (−4.58 − 2.44i)27-s + (−6.85 − 3.95i)29-s + (−1.78 + 1.02i)31-s + ⋯
L(s)  = 1  + (0.352 + 0.935i)3-s + (0.857 + 1.48i)5-s + (0.914 − 0.404i)7-s + (−0.751 + 0.659i)9-s + (0.176 + 0.101i)11-s + (1.17 − 0.675i)13-s + (−1.08 + 1.32i)15-s − 1.40·17-s − 0.969i·19-s + (0.700 + 0.713i)21-s + (0.993 − 0.573i)23-s + (−0.972 + 1.68i)25-s + (−0.882 − 0.471i)27-s + (−1.27 − 0.734i)29-s + (−0.320 + 0.184i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0850 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0850 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.0850 - 0.996i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.0850 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44695 + 1.32864i\)
\(L(\frac12)\) \(\approx\) \(1.44695 + 1.32864i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.610 - 1.62i)T \)
7 \( 1 + (-2.41 + 1.06i)T \)
good5 \( 1 + (-1.91 - 3.32i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.585 - 0.338i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.22 + 2.43i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.79T + 17T^{2} \)
19 \( 1 + 4.22iT - 19T^{2} \)
23 \( 1 + (-4.76 + 2.75i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (6.85 + 3.95i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.78 - 1.02i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 8.71T + 37T^{2} \)
41 \( 1 + (-4.84 - 8.38i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.57 + 6.19i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.666 - 1.15i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 5.18iT - 53T^{2} \)
59 \( 1 + (-2.09 - 3.63i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.38 - 1.37i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.27 + 5.67i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.1iT - 71T^{2} \)
73 \( 1 + 3.65iT - 73T^{2} \)
79 \( 1 + (5.61 - 9.72i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.61 + 7.98i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 1.79T + 89T^{2} \)
97 \( 1 + (2.60 + 1.50i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.78614706782840584300844764483, −10.63114541828116685251160804284, −9.371635443434018748361434867448, −8.673907920620126592387445284362, −7.47247918818149902538751092983, −6.51937029647176786254700749064, −5.48977046128592736696968395413, −4.33303632928574287765806299606, −3.19720956024686090086892003519, −2.14116462680716737242884084277, 1.36963547004584747844869686611, 1.99720292225785422290663023668, 3.95069703011531963364081395676, 5.25798055842579743302525728518, 5.93532889172649052672334343331, 7.08170425634616017448550515224, 8.329059698290699317437634273108, 8.853126203464542837223927498324, 9.300146008334096779057080248772, 10.96776912555660322952457622232

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