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Properties

Label 2-504-504.355-c1-0-80
Degree $2$
Conductor $504$
Sign $-0.520 + 0.853i$
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 − i)2-s + 1.73i·3-s − 2i·4-s + (−0.866 − 1.5i)5-s + (1.73 + 1.73i)6-s + (−1.73 − 2i)7-s + (−2 − 2i)8-s − 2.99·9-s + (−2.36 − 0.633i)10-s + (2.5 − 4.33i)11-s + 3.46·12-s + (−0.866 + 1.5i)13-s + (−3.73 − 0.267i)14-s + (2.59 − 1.49i)15-s − 4·16-s + (1.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s + 0.999i·3-s − i·4-s + (−0.387 − 0.670i)5-s + (0.707 + 0.707i)6-s + (−0.654 − 0.755i)7-s + (−0.707 − 0.707i)8-s − 0.999·9-s + (−0.748 − 0.200i)10-s + (0.753 − 1.30i)11-s + 1.00·12-s + (−0.240 + 0.416i)13-s + (−0.997 − 0.0716i)14-s + (0.670 − 0.387i)15-s − 16-s + (0.363 − 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.520 + 0.853i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.520 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.520 + 0.853i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (355, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.520 + 0.853i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.724043 - 1.28940i\)
\(L(\frac12)\) \(\approx\) \(0.724043 - 1.28940i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
3 \( 1 - 1.73iT \)
7 \( 1 + (1.73 + 2i)T \)
good5 \( 1 + (0.866 + 1.5i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.5 + 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.866 - 1.5i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.5 + 0.866i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.5 + 0.866i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.06 - 3.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-6.06 + 3.5i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.92T + 31T^{2} \)
37 \( 1 + (0.866 + 0.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-10.5 - 6.06i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.5 + 7.79i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.46T + 47T^{2} \)
53 \( 1 + (-9.52 + 5.5i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 6.92iT - 59T^{2} \)
61 \( 1 + 3.46T + 61T^{2} \)
67 \( 1 + 8T + 67T^{2} \)
71 \( 1 - 10iT - 71T^{2} \)
73 \( 1 + (7.5 - 4.33i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 6iT - 79T^{2} \)
83 \( 1 + (-13.5 + 7.79i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.5 - 4.33i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.5 + 0.866i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.58469735495602496522593579631, −9.952524423308190468903987825497, −9.127691906393613551824657549623, −8.231944208967156654316478293042, −6.53310558806318440545704429545, −5.72991353229310829738351881843, −4.44977000388254875768371618797, −3.95574820829292894217575661107, −2.91737771817043246453735946834, −0.70012247849513531544941636128, 2.31304894017500441067379821647, 3.30530371173192085313716048738, 4.66201839809633003629275355393, 6.04266861940329737826507298603, 6.54299872048024963407176290608, 7.38090039808502827257865916177, 8.172261251449355891314383585271, 9.164655508807905146282543133654, 10.40597652513523957253361902331, 11.80148186856340096776050181752

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