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Properties

Label 2-504-9.4-c1-0-11
Degree $2$
Conductor $504$
Sign $-0.613 + 0.789i$
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  + (−1.58 − 0.702i)3-s + (−1.53 + 2.65i)5-s + (0.5 + 0.866i)7-s + (2.01 + 2.22i)9-s + (−2.86 − 4.96i)11-s + (1.44 − 2.50i)13-s + (4.30 − 3.13i)15-s − 4.02·17-s − 0.899·19-s + (−0.183 − 1.72i)21-s + (1.86 − 3.22i)23-s + (−2.21 − 3.83i)25-s + (−1.62 − 4.93i)27-s + (−4.43 − 7.67i)29-s + (0.651 − 1.12i)31-s + ⋯
L(s)  = 1  + (−0.914 − 0.405i)3-s + (−0.686 + 1.18i)5-s + (0.188 + 0.327i)7-s + (0.670 + 0.741i)9-s + (−0.864 − 1.49i)11-s + (0.401 − 0.695i)13-s + (1.11 − 0.808i)15-s − 0.976·17-s − 0.206·19-s + (−0.0399 − 0.375i)21-s + (0.388 − 0.672i)23-s + (−0.443 − 0.767i)25-s + (−0.312 − 0.949i)27-s + (−0.823 − 1.42i)29-s + (0.116 − 0.202i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.613 + 0.789i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.613 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.613 + 0.789i$
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: \((2,\ 504,\ (\ :1/2),\ -0.613 + 0.789i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.152975 - 0.312666i\)
\(L(\frac12)\) \(\approx\) \(0.152975 - 0.312666i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.58 + 0.702i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (1.53 - 2.65i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.86 + 4.96i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.44 + 2.50i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.02T + 17T^{2} \)
19 \( 1 + 0.899T + 19T^{2} \)
23 \( 1 + (-1.86 + 3.22i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.43 + 7.67i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.651 + 1.12i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 3.56T + 37T^{2} \)
41 \( 1 + (4.99 - 8.65i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.55 + 4.41i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.40 + 9.35i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.43T + 53T^{2} \)
59 \( 1 + (2.79 - 4.84i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.33 + 4.04i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.21 + 7.30i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.17T + 71T^{2} \)
73 \( 1 - 8.54T + 73T^{2} \)
79 \( 1 + (-0.689 - 1.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.44 - 11.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 5.27T + 89T^{2} \)
97 \( 1 + (8.60 + 14.9i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.99315888319897539559781262044, −10.14093514109327206483636581965, −8.481754885061508664032654815927, −7.86993650873792857138805078503, −6.79161082411813644467270813034, −6.07905755219719510573092443252, −5.11133888040514925132535101308, −3.64958759302396496811489661224, −2.47411213579881936206042819291, −0.23281199961062719436523020068, 1.59281154424071540172458661197, 3.88039143924601376866018371171, 4.72197195736060383083108762239, 5.22890199059903767654855869268, 6.74701969123503566182527382711, 7.49717776718483712421777795182, 8.710839842063387753677366621868, 9.450123266228713420125453795406, 10.49088333701364848803069679509, 11.23402984904416005041649992926

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