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Properties

Label 2-504-56.3-c1-0-12
Degree $2$
Conductor $504$
Sign $-0.350 - 0.936i$
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 + (2.08 + 3.61i)5-s + (2.39 + 1.12i)7-s + (2.79 − 0.407i)8-s + (−5.89 + 0.284i)10-s + (0.855 − 1.48i)11-s + 1.54·13-s + (−2.95 + 2.28i)14-s + (−1.29 + 3.78i)16-s + (−2.02 − 1.16i)17-s + (6.09 − 3.52i)19-s + (3.45 − 7.60i)20-s + (1.30 + 2.03i)22-s + (−0.406 + 0.234i)23-s + ⋯
L(s)  = 1  + (−0.457 + 0.889i)2-s + (−0.581 − 0.813i)4-s + (0.933 + 1.61i)5-s + (0.905 + 0.423i)7-s + (0.989 − 0.144i)8-s + (−1.86 + 0.0900i)10-s + (0.257 − 0.446i)11-s + 0.427·13-s + (−0.791 + 0.611i)14-s + (−0.324 + 0.945i)16-s + (−0.490 − 0.282i)17-s + (1.39 − 0.807i)19-s + (0.773 − 1.69i)20-s + (0.279 + 0.433i)22-s + (−0.0846 + 0.0488i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.350 - 0.936i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.350 - 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.804630 + 1.15998i\)
\(L(\frac12)\) \(\approx\) \(0.804630 + 1.15998i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.39 - 1.12i)T \)
good5 \( 1 + (-2.08 - 3.61i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.855 + 1.48i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.54T + 13T^{2} \)
17 \( 1 + (2.02 + 1.16i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.09 + 3.52i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.406 - 0.234i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.33iT - 29T^{2} \)
31 \( 1 + (1.58 - 2.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (7.74 - 4.47i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.31iT - 41T^{2} \)
43 \( 1 + 3.42T + 43T^{2} \)
47 \( 1 + (2.95 + 5.11i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.35 + 0.781i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.26 - 3.04i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.55 + 7.89i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.73 + 6.47i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.49iT - 71T^{2} \)
73 \( 1 + (12.5 + 7.26i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.46 + 0.843i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.72iT - 83T^{2} \)
89 \( 1 + (1.83 - 1.06i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.95iT - 97T^{2} \)
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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

−11.05501294033617788416528697658, −10.17733351999887450172953218492, −9.388176760527018448975426287682, −8.480586330130082368042050868640, −7.39198995451516505058971037637, −6.67555342558645412124847802969, −5.84544698419857840126567535826, −4.96008206756391546262005143942, −3.19100946404962523184872122424, −1.75749920762677680674845130967, 1.17600325415290934755067588586, 1.91961204313311942822625500481, 3.85967093329045248127849527981, 4.81104157731936532383485387198, 5.61007322212286131734104129344, 7.34150922015734607143000186984, 8.349580940684098685898382589525, 8.946332489739623480616116241513, 9.747938458699394901626379246568, 10.51471310045591256552239836738

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