Properties

Label 2-504-56.19-c1-0-37
Degree $2$
Conductor $504$
Sign $-0.967 + 0.253i$
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  + (0.647 − 1.25i)2-s + (−1.16 − 1.62i)4-s + (1.61 − 2.79i)5-s + (−1.82 − 1.91i)7-s + (−2.79 + 0.406i)8-s + (−2.46 − 3.83i)10-s + (1.10 + 1.91i)11-s − 5.08·13-s + (−3.58 + 1.05i)14-s + (−1.30 + 3.78i)16-s + (2.73 − 1.57i)17-s + (2.93 + 1.69i)19-s + (−6.42 + 0.619i)20-s + (3.12 − 0.150i)22-s + (2.65 + 1.53i)23-s + ⋯
L(s)  = 1  + (0.457 − 0.889i)2-s + (−0.580 − 0.814i)4-s + (0.721 − 1.25i)5-s + (−0.690 − 0.723i)7-s + (−0.989 + 0.143i)8-s + (−0.781 − 1.21i)10-s + (0.333 + 0.577i)11-s − 1.40·13-s + (−0.959 + 0.282i)14-s + (−0.325 + 0.945i)16-s + (0.663 − 0.383i)17-s + (0.673 + 0.388i)19-s + (−1.43 + 0.138i)20-s + (0.666 − 0.0320i)22-s + (0.553 + 0.319i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 + 0.253i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.967 + 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.967 + 0.253i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.967 + 0.253i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.197946 - 1.53629i\)
\(L(\frac12)\) \(\approx\) \(0.197946 - 1.53629i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.647 + 1.25i)T \)
3 \( 1 \)
7 \( 1 + (1.82 + 1.91i)T \)
good5 \( 1 + (-1.61 + 2.79i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.10 - 1.91i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.08T + 13T^{2} \)
17 \( 1 + (-2.73 + 1.57i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.93 - 1.69i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.65 - 1.53i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 9.88iT - 29T^{2} \)
31 \( 1 + (-1.01 - 1.75i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.798 - 0.460i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.96iT - 41T^{2} \)
43 \( 1 + 6.68T + 43T^{2} \)
47 \( 1 + (-1.06 + 1.83i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.12 - 1.80i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-10.6 + 6.14i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.34 + 10.9i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.40 + 7.63i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.7iT - 71T^{2} \)
73 \( 1 + (-7.82 + 4.51i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-9.60 - 5.54i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.57iT - 83T^{2} \)
89 \( 1 + (-6.32 - 3.65i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 15.2iT - 97T^{2} \)
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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

−10.25227134023415810594033811764, −9.662946524759091844302314934323, −9.335237045207137256128756763929, −7.893343400069197290249402275794, −6.65124187574392058953671433666, −5.39724756526832469510516941242, −4.79265070496730128949071990626, −3.64492631071291481730084887055, −2.19818811508485868717146684782, −0.810286763537858294378261309042, 2.68966876385712586341742077031, 3.33969024735529551636311375908, 5.05146155825305377441297258338, 5.87838233124788975712533681383, 6.72307322752386428413068978125, 7.31407262595123056186027445047, 8.617081857112586431143137938340, 9.527067037773706890770288973549, 10.20431002833091290975642561230, 11.47107585903084360008871955833

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