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Properties

Label 2-504-168.11-c1-0-7
Degree $2$
Conductor $504$
Sign $0.999 + 0.0294i$
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  + (−1.36 − 0.366i)2-s + (1.73 + i)4-s + (−0.914 − 1.58i)5-s + (−0.358 + 2.62i)7-s + (−1.99 − 2i)8-s + (0.669 + 2.49i)10-s + (1.37 + 0.792i)11-s − 2.58i·13-s + (1.44 − 3.44i)14-s + (1.99 + 3.46i)16-s + (5.40 + 3.12i)17-s + (1.29 + 2.23i)19-s − 3.65i·20-s + (−1.58 − 1.58i)22-s + (−0.292 − 0.507i)23-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (0.866 + 0.5i)4-s + (−0.408 − 0.708i)5-s + (−0.135 + 0.990i)7-s + (−0.707 − 0.707i)8-s + (0.211 + 0.789i)10-s + (0.414 + 0.239i)11-s − 0.717i·13-s + (0.387 − 0.921i)14-s + (0.499 + 0.866i)16-s + (1.31 + 0.757i)17-s + (0.296 + 0.513i)19-s − 0.817i·20-s + (−0.338 − 0.338i)22-s + (−0.0610 − 0.105i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0294i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0294i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.999 + 0.0294i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.999 + 0.0294i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.889938 - 0.0131092i\)
\(L(\frac12)\) \(\approx\) \(0.889938 - 0.0131092i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.366i)T \)
3 \( 1 \)
7 \( 1 + (0.358 - 2.62i)T \)
good5 \( 1 + (0.914 + 1.58i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.37 - 0.792i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.58iT - 13T^{2} \)
17 \( 1 + (-5.40 - 3.12i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.29 - 2.23i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.292 + 0.507i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + (-3.82 - 2.20i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-7.85 + 4.53i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 8.82iT - 41T^{2} \)
43 \( 1 - 11.4T + 43T^{2} \)
47 \( 1 + (-1.29 - 2.23i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.08 + 5.34i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.98 - 4.03i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.40 - 3.12i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.82 + 6.63i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 13.8T + 71T^{2} \)
73 \( 1 + (-3.65 + 6.33i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.98 - 4.03i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 8.07iT - 83T^{2} \)
89 \( 1 + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 15.8T + 97T^{2} \)
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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

−10.82375927494076679051280126985, −9.839298554854122045631022957007, −9.190715695772189774822922838969, −8.192780793667742867401212626497, −7.79383254849754025931478007156, −6.34286608185081323633014511484, −5.47078684722452037230447078049, −3.91139849776864281328854031546, −2.67240497087186932065686863757, −1.12798675272250650313428973648, 0.958997109696094029357660239611, 2.78963463118252470413080130280, 3.98930624532017902195333376167, 5.56715964427855305053985959141, 6.78244469552851994238585639437, 7.27039162273149935226490696118, 8.070158477188617998830777363599, 9.336279035732783926594658447607, 9.878192221156349792272007490854, 10.95042964363033592416546710785

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