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Properties

Label 2-504-63.38-c1-0-8
Degree $2$
Conductor $504$
Sign $-0.687 - 0.726i$
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  + (0.957 + 1.44i)3-s + (−1.80 + 3.12i)5-s + (2.05 − 1.66i)7-s + (−1.16 + 2.76i)9-s + (−4.30 + 2.48i)11-s + (−0.167 + 0.0966i)13-s + (−6.23 + 0.387i)15-s + (0.257 − 0.446i)17-s + (1.69 − 0.979i)19-s + (4.37 + 1.37i)21-s + (4.75 + 2.74i)23-s + (−4.00 − 6.94i)25-s + (−5.10 + 0.960i)27-s + (−6.81 − 3.93i)29-s + 3.21i·31-s + ⋯
L(s)  = 1  + (0.552 + 0.833i)3-s + (−0.806 + 1.39i)5-s + (0.776 − 0.629i)7-s + (−0.389 + 0.921i)9-s + (−1.29 + 0.749i)11-s + (−0.0464 + 0.0267i)13-s + (−1.61 + 0.0999i)15-s + (0.0625 − 0.108i)17-s + (0.389 − 0.224i)19-s + (0.954 + 0.299i)21-s + (0.992 + 0.572i)23-s + (−0.801 − 1.38i)25-s + (−0.982 + 0.184i)27-s + (−1.26 − 0.730i)29-s + 0.578i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.687 - 0.726i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.687 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.687 - 0.726i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.687 - 0.726i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.518409 + 1.20382i\)
\(L(\frac12)\) \(\approx\) \(0.518409 + 1.20382i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.957 - 1.44i)T \)
7 \( 1 + (-2.05 + 1.66i)T \)
good5 \( 1 + (1.80 - 3.12i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.30 - 2.48i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.167 - 0.0966i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.257 + 0.446i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.69 + 0.979i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.75 - 2.74i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (6.81 + 3.93i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.21iT - 31T^{2} \)
37 \( 1 + (-4.09 - 7.10i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.39 + 5.88i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.07 - 8.78i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 11.5T + 47T^{2} \)
53 \( 1 + (-6.88 - 3.97i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 9.61T + 59T^{2} \)
61 \( 1 + 0.692iT - 61T^{2} \)
67 \( 1 + 2.52T + 67T^{2} \)
71 \( 1 - 13.8iT - 71T^{2} \)
73 \( 1 + (-5.49 - 3.17i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 7.98T + 79T^{2} \)
83 \( 1 + (-4.86 + 8.42i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.42 + 7.65i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.49 - 1.43i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.07142934323408639605923772450, −10.42288688551624016821641510715, −9.754628734334592529902802409050, −8.403141292124776648247039170857, −7.52632473909734362295860183514, −7.17520212897548809832158463305, −5.37593603182106815690736059113, −4.41751809042763558947717865215, −3.40217698130728157542680118590, −2.43432097762642323175747450116, 0.73681264109124833724667545896, 2.24472009599952562662524692203, 3.64802516496597706631979432355, 5.04055742008424855106306638174, 5.68754846099481063810994226600, 7.34504333002837120283331270591, 8.020813503944898255251091355474, 8.618349420562310410507537826392, 9.228840198235718173814327109848, 10.83371184981258721265823419179

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