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Properties

Label 2-552-23.3-c1-0-0
Degree $2$
Conductor $552$
Sign $-0.633 + 0.774i$
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 + (−2.29 + 2.65i)5-s + (−2.05 + 1.32i)7-s + (−0.654 − 0.755i)9-s + (0.516 − 3.59i)11-s + (−1.06 − 0.686i)13-s + (−1.45 − 3.19i)15-s + (−3.04 − 0.893i)17-s + (5.42 − 1.59i)19-s + (−0.348 − 2.42i)21-s + (−2.27 − 4.22i)23-s + (−1.04 − 7.24i)25-s + (0.959 − 0.281i)27-s + (−3.58 − 1.05i)29-s + (1.20 + 2.63i)31-s + ⋯
L(s)  = 1  + (−0.239 + 0.525i)3-s + (−1.02 + 1.18i)5-s + (−0.778 + 0.500i)7-s + (−0.218 − 0.251i)9-s + (0.155 − 1.08i)11-s + (−0.296 − 0.190i)13-s + (−0.376 − 0.824i)15-s + (−0.738 − 0.216i)17-s + (1.24 − 0.365i)19-s + (−0.0759 − 0.528i)21-s + (−0.473 − 0.880i)23-s + (−0.208 − 1.44i)25-s + (0.184 − 0.0542i)27-s + (−0.666 − 0.195i)29-s + (0.216 + 0.473i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.633 + 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.633 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.633 + 0.774i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ -0.633 + 0.774i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0164373 - 0.0346767i\)
\(L(\frac12)\) \(\approx\) \(0.0164373 - 0.0346767i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.27 + 4.22i)T \)
good5 \( 1 + (2.29 - 2.65i)T + (-0.711 - 4.94i)T^{2} \)
7 \( 1 + (2.05 - 1.32i)T + (2.90 - 6.36i)T^{2} \)
11 \( 1 + (-0.516 + 3.59i)T + (-10.5 - 3.09i)T^{2} \)
13 \( 1 + (1.06 + 0.686i)T + (5.40 + 11.8i)T^{2} \)
17 \( 1 + (3.04 + 0.893i)T + (14.3 + 9.19i)T^{2} \)
19 \( 1 + (-5.42 + 1.59i)T + (15.9 - 10.2i)T^{2} \)
29 \( 1 + (3.58 + 1.05i)T + (24.3 + 15.6i)T^{2} \)
31 \( 1 + (-1.20 - 2.63i)T + (-20.3 + 23.4i)T^{2} \)
37 \( 1 + (-0.367 - 0.423i)T + (-5.26 + 36.6i)T^{2} \)
41 \( 1 + (6.77 - 7.82i)T + (-5.83 - 40.5i)T^{2} \)
43 \( 1 + (-1.22 + 2.68i)T + (-28.1 - 32.4i)T^{2} \)
47 \( 1 + 11.8T + 47T^{2} \)
53 \( 1 + (9.81 - 6.30i)T + (22.0 - 48.2i)T^{2} \)
59 \( 1 + (-9.75 - 6.26i)T + (24.5 + 53.6i)T^{2} \)
61 \( 1 + (2.34 + 5.13i)T + (-39.9 + 46.1i)T^{2} \)
67 \( 1 + (1.66 + 11.5i)T + (-64.2 + 18.8i)T^{2} \)
71 \( 1 + (0.110 + 0.765i)T + (-68.1 + 20.0i)T^{2} \)
73 \( 1 + (13.7 - 4.03i)T + (61.4 - 39.4i)T^{2} \)
79 \( 1 + (5.98 + 3.84i)T + (32.8 + 71.8i)T^{2} \)
83 \( 1 + (2.61 + 3.01i)T + (-11.8 + 82.1i)T^{2} \)
89 \( 1 + (-3.55 + 7.78i)T + (-58.2 - 67.2i)T^{2} \)
97 \( 1 + (5.63 - 6.50i)T + (-13.8 - 96.0i)T^{2} \)
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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

−11.42524969801665346670790207328, −10.56189952610745344754387832516, −9.712301698421783344402466452843, −8.748843037401794004235611525599, −7.75153742407435308821721174964, −6.72229149411461596590678058946, −6.03283554812211517781679011941, −4.70650921831750365926718219677, −3.40370922016123528176231993668, −2.92459697124135534117573158812, 0.02254676722700840004496771003, 1.62190016875246665532194479776, 3.54370239015416163229679143327, 4.48183313236400676355232308483, 5.45700921812939303330341637094, 6.82787795369505467171754865285, 7.47137354567182211477859357211, 8.308338298396422158782120207512, 9.389348045722380558235637547496, 10.04491958914749048110510782846

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