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Properties

Label 2-552-23.6-c1-0-7
Degree $2$
Conductor $552$
Sign $-0.337 + 0.941i$
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  + (−0.959 − 0.281i)3-s + (−2.20 + 1.41i)5-s + (−0.514 + 3.58i)7-s + (0.841 + 0.540i)9-s + (−2.22 − 4.87i)11-s + (−0.416 − 2.89i)13-s + (2.51 − 0.737i)15-s + (5.05 − 5.83i)17-s + (−0.423 − 0.488i)19-s + (1.50 − 3.29i)21-s + (−0.961 − 4.69i)23-s + (0.769 − 1.68i)25-s + (−0.654 − 0.755i)27-s + (−2.40 + 2.77i)29-s + (−5.05 + 1.48i)31-s + ⋯
L(s)  = 1  + (−0.553 − 0.162i)3-s + (−0.984 + 0.632i)5-s + (−0.194 + 1.35i)7-s + (0.280 + 0.180i)9-s + (−0.671 − 1.47i)11-s + (−0.115 − 0.802i)13-s + (0.648 − 0.190i)15-s + (1.22 − 1.41i)17-s + (−0.0971 − 0.112i)19-s + (0.328 − 0.718i)21-s + (−0.200 − 0.979i)23-s + (0.153 − 0.337i)25-s + (−0.126 − 0.145i)27-s + (−0.446 + 0.515i)29-s + (−0.907 + 0.266i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.337 + 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.337 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.337 + 0.941i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ -0.337 + 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.254213 - 0.361233i\)
\(L(\frac12)\) \(\approx\) \(0.254213 - 0.361233i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.959 + 0.281i)T \)
23 \( 1 + (0.961 + 4.69i)T \)
good5 \( 1 + (2.20 - 1.41i)T + (2.07 - 4.54i)T^{2} \)
7 \( 1 + (0.514 - 3.58i)T + (-6.71 - 1.97i)T^{2} \)
11 \( 1 + (2.22 + 4.87i)T + (-7.20 + 8.31i)T^{2} \)
13 \( 1 + (0.416 + 2.89i)T + (-12.4 + 3.66i)T^{2} \)
17 \( 1 + (-5.05 + 5.83i)T + (-2.41 - 16.8i)T^{2} \)
19 \( 1 + (0.423 + 0.488i)T + (-2.70 + 18.8i)T^{2} \)
29 \( 1 + (2.40 - 2.77i)T + (-4.12 - 28.7i)T^{2} \)
31 \( 1 + (5.05 - 1.48i)T + (26.0 - 16.7i)T^{2} \)
37 \( 1 + (2.76 + 1.77i)T + (15.3 + 33.6i)T^{2} \)
41 \( 1 + (-10.4 + 6.70i)T + (17.0 - 37.2i)T^{2} \)
43 \( 1 + (6.82 + 2.00i)T + (36.1 + 23.2i)T^{2} \)
47 \( 1 + 10.1T + 47T^{2} \)
53 \( 1 + (0.257 - 1.78i)T + (-50.8 - 14.9i)T^{2} \)
59 \( 1 + (-0.444 - 3.08i)T + (-56.6 + 16.6i)T^{2} \)
61 \( 1 + (-5.55 + 1.63i)T + (51.3 - 32.9i)T^{2} \)
67 \( 1 + (-5.49 + 12.0i)T + (-43.8 - 50.6i)T^{2} \)
71 \( 1 + (-1.62 + 3.56i)T + (-46.4 - 53.6i)T^{2} \)
73 \( 1 + (-4.29 - 4.95i)T + (-10.3 + 72.2i)T^{2} \)
79 \( 1 + (-1.44 - 10.0i)T + (-75.7 + 22.2i)T^{2} \)
83 \( 1 + (2.72 + 1.75i)T + (34.4 + 75.4i)T^{2} \)
89 \( 1 + (12.3 + 3.63i)T + (74.8 + 48.1i)T^{2} \)
97 \( 1 + (7.03 - 4.52i)T + (40.2 - 88.2i)T^{2} \)
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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

−10.88351051736937395436966162802, −9.698117765145480448881491524691, −8.576663394662761006921647972390, −7.84994422193644617046426226314, −6.92366793241536087852629172531, −5.68895683693835005522515216648, −5.26689563052849501428411564252, −3.45757263271436020317727885845, −2.72053274014652747138224914732, −0.28125984242662832349062600526, 1.49692910152501995811444474938, 3.78865014515493978512310547548, 4.27637770997671732827148959443, 5.30953702879633456064944952028, 6.66756025035883733167583638468, 7.59083717898238815547239138675, 8.052283881204790688612507102346, 9.646960944779212672075527403310, 10.09165553347851180044741974866, 11.09130464904378982506101897931

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