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Properties

Label 2-552-23.8-c1-0-7
Degree $2$
Conductor $552$
Sign $0.894 + 0.447i$
Analytic cond. $4.40774$
Root an. cond. $2.09946$
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  + (0.415 + 0.909i)3-s + (−1.20 − 1.39i)5-s + (−0.921 − 0.592i)7-s + (−0.654 + 0.755i)9-s + (−0.117 − 0.814i)11-s + (4.25 − 2.73i)13-s + (0.764 − 1.67i)15-s + (6.77 − 1.98i)17-s + (6.40 + 1.88i)19-s + (0.155 − 1.08i)21-s + (0.349 − 4.78i)23-s + (0.229 − 1.59i)25-s + (−0.959 − 0.281i)27-s + (−7.90 + 2.32i)29-s + (−3.03 + 6.64i)31-s + ⋯
L(s)  = 1  + (0.239 + 0.525i)3-s + (−0.538 − 0.621i)5-s + (−0.348 − 0.223i)7-s + (−0.218 + 0.251i)9-s + (−0.0352 − 0.245i)11-s + (1.18 − 0.758i)13-s + (0.197 − 0.432i)15-s + (1.64 − 0.482i)17-s + (1.46 + 0.431i)19-s + (0.0340 − 0.236i)21-s + (0.0728 − 0.997i)23-s + (0.0459 − 0.319i)25-s + (−0.184 − 0.0542i)27-s + (−1.46 + 0.431i)29-s + (−0.544 + 1.19i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40862 - 0.333180i\)
\(L(\frac12)\) \(\approx\) \(1.40862 - 0.333180i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.415 - 0.909i)T \)
23 \( 1 + (-0.349 + 4.78i)T \)
good5 \( 1 + (1.20 + 1.39i)T + (-0.711 + 4.94i)T^{2} \)
7 \( 1 + (0.921 + 0.592i)T + (2.90 + 6.36i)T^{2} \)
11 \( 1 + (0.117 + 0.814i)T + (-10.5 + 3.09i)T^{2} \)
13 \( 1 + (-4.25 + 2.73i)T + (5.40 - 11.8i)T^{2} \)
17 \( 1 + (-6.77 + 1.98i)T + (14.3 - 9.19i)T^{2} \)
19 \( 1 + (-6.40 - 1.88i)T + (15.9 + 10.2i)T^{2} \)
29 \( 1 + (7.90 - 2.32i)T + (24.3 - 15.6i)T^{2} \)
31 \( 1 + (3.03 - 6.64i)T + (-20.3 - 23.4i)T^{2} \)
37 \( 1 + (-6.75 + 7.79i)T + (-5.26 - 36.6i)T^{2} \)
41 \( 1 + (1.36 + 1.57i)T + (-5.83 + 40.5i)T^{2} \)
43 \( 1 + (-2.23 - 4.88i)T + (-28.1 + 32.4i)T^{2} \)
47 \( 1 - 0.433T + 47T^{2} \)
53 \( 1 + (5.96 + 3.83i)T + (22.0 + 48.2i)T^{2} \)
59 \( 1 + (-4.38 + 2.81i)T + (24.5 - 53.6i)T^{2} \)
61 \( 1 + (4.75 - 10.4i)T + (-39.9 - 46.1i)T^{2} \)
67 \( 1 + (1.32 - 9.24i)T + (-64.2 - 18.8i)T^{2} \)
71 \( 1 + (-0.797 + 5.54i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (-3.51 - 1.03i)T + (61.4 + 39.4i)T^{2} \)
79 \( 1 + (-6.05 + 3.89i)T + (32.8 - 71.8i)T^{2} \)
83 \( 1 + (6.97 - 8.05i)T + (-11.8 - 82.1i)T^{2} \)
89 \( 1 + (3.91 + 8.56i)T + (-58.2 + 67.2i)T^{2} \)
97 \( 1 + (-4.70 - 5.42i)T + (-13.8 + 96.0i)T^{2} \)
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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

−10.65251554985230752473036981683, −9.826073598616793723463814946047, −8.961366132103689934522459342553, −8.100554155037562929118818761220, −7.38825817816689320275325050924, −5.87652062872099254310339036125, −5.14487021158484822983644701795, −3.80529778959176764064086772094, −3.17120655997186746565810247885, −0.965136749261919110533426326346, 1.46017922851685276557252817556, 3.12347113735994546113697971478, 3.79889944343069548405840768446, 5.50616480685418595685397167122, 6.34083932822884316564581106478, 7.50010146042809899515779525843, 7.83072554985666418576550520664, 9.225462595822826095010499831052, 9.761558991706364806337392332862, 11.21116907118509104809887267052

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