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Properties

Label 2-504-56.37-c1-0-1
Degree $2$
Conductor $504$
Sign $-0.980 + 0.198i$
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  + (1.22 + 0.707i)2-s + (0.999 + 1.73i)4-s + (−3.82 − 2.20i)5-s + (−2.62 + 0.358i)7-s + 2.82i·8-s + (−3.12 − 5.40i)10-s + (−5.04 + 2.91i)11-s + (−3.46 − 1.41i)14-s + (−2.00 + 3.46i)16-s − 8.82i·20-s − 8.24·22-s + (7.24 + 12.5i)25-s + (−3.24 − 4.18i)28-s − 7.58i·29-s + (−0.378 − 0.655i)31-s + (−4.89 + 2.82i)32-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)2-s + (0.499 + 0.866i)4-s + (−1.70 − 0.987i)5-s + (−0.990 + 0.135i)7-s + 0.999i·8-s + (−0.987 − 1.70i)10-s + (−1.52 + 0.878i)11-s + (−0.925 − 0.377i)14-s + (−0.500 + 0.866i)16-s − 1.97i·20-s − 1.75·22-s + (1.44 + 2.50i)25-s + (−0.612 − 0.790i)28-s − 1.40i·29-s + (−0.0680 − 0.117i)31-s + (−0.866 + 0.499i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.198i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 + 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.980 + 0.198i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.980 + 0.198i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0332449 - 0.332382i\)
\(L(\frac12)\) \(\approx\) \(0.0332449 - 0.332382i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 - 0.707i)T \)
3 \( 1 \)
7 \( 1 + (2.62 - 0.358i)T \)
good5 \( 1 + (3.82 + 2.20i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (5.04 - 2.91i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.58iT - 29T^{2} \)
31 \( 1 + (0.378 + 0.655i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.52 - 2.03i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.89 - 1.67i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (7 + 12.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.86 - 15.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12.1iT - 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 17.9T + 97T^{2} \)
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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

−11.76226125872886005558344169998, −10.75958835362566212617400322169, −9.468984867324106230187210781376, −8.295032046750335848700924674853, −7.74372541652705191014184922232, −6.96148212405879362967414927338, −5.59909080930352825590496823391, −4.67948044863502671991038598619, −3.89972900788846604767602282237, −2.74774382084406092953052371020, 0.14016251005132513698023917893, 2.93362571980390290213624692251, 3.27141121622401510867074093909, 4.39983156338771042715490629319, 5.67864350040100179506955387195, 6.78940226118516075849565598044, 7.45966158337519045211612504908, 8.532862801159932028188235329679, 10.05117957638941729038414960136, 10.77094226278805963851364815379

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