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Properties

Label 2-504-7.2-c1-0-4
Degree $2$
Conductor $504$
Sign $0.991 + 0.126i$
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  + (−0.5 + 0.866i)5-s + (0.5 − 2.59i)7-s + (1.5 + 2.59i)11-s + 4·13-s + (2 − 3.46i)19-s + (4 − 6.92i)23-s + (2 + 3.46i)25-s + 3·29-s + (2.5 + 4.33i)31-s + (2 + 1.73i)35-s + (−4 + 6.92i)37-s − 8·41-s + 6·43-s + (5 − 8.66i)47-s + (−6.5 − 2.59i)49-s + ⋯
L(s)  = 1  + (−0.223 + 0.387i)5-s + (0.188 − 0.981i)7-s + (0.452 + 0.783i)11-s + 1.10·13-s + (0.458 − 0.794i)19-s + (0.834 − 1.44i)23-s + (0.400 + 0.692i)25-s + 0.557·29-s + (0.449 + 0.777i)31-s + (0.338 + 0.292i)35-s + (−0.657 + 1.13i)37-s − 1.24·41-s + 0.914·43-s + (0.729 − 1.26i)47-s + (−0.928 − 0.371i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.991 + 0.126i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50702 - 0.0956360i\)
\(L(\frac12)\) \(\approx\) \(1.50702 - 0.0956360i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (-0.5 + 2.59i)T \)
good5 \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4 + 6.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + (-2.5 - 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4 - 6.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 - 6T + 43T^{2} \)
47 \( 1 + (-5 + 8.66i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.5 - 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.5 + 4.33i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3 + 5.19i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10T + 71T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.5 - 9.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7T + 83T^{2} \)
89 \( 1 + (9 - 15.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 17T + 97T^{2} \)
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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

−10.79456251038982872073303338742, −10.25835875651490948168694898395, −9.065068956014744154161589686847, −8.236893518893041840682346735070, −6.96023274190169157949048044455, −6.72930363099288954951121516728, −5.07057680900019599616935491657, −4.13431469008601160659979963729, −3.02037430562676103778859869525, −1.21089169785677624677076404699, 1.32712485904019117575690321372, 3.01535134390573336343941041426, 4.12653489596136534809295839707, 5.50403315614617341918933317591, 6.06227841437955689729721380516, 7.39915350766115575884578249705, 8.573070825867855166115179036695, 8.840257192656848746535689843757, 10.00263326156933108535713621077, 11.16629336353761874326569095686

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