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Properties

Label 2-552-69.68-c1-0-22
Degree $2$
Conductor $552$
Sign $-0.992 + 0.119i$
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  + (0.618 − 1.61i)3-s − 2.28·5-s − 1.41i·7-s + (−2.23 − 2.00i)9-s − 1.95·11-s + 1.23·13-s + (−1.41 + 3.70i)15-s − 7.73·17-s + 1.20i·19-s + (−2.28 − 0.874i)21-s + (4.24 − 2.23i)23-s + 0.236·25-s + (−4.61 + 2.38i)27-s − 0.763i·29-s − 6·31-s + ⋯
L(s)  = 1  + (0.356 − 0.934i)3-s − 1.02·5-s − 0.534i·7-s + (−0.745 − 0.666i)9-s − 0.589·11-s + 0.342·13-s + (−0.365 + 0.955i)15-s − 1.87·17-s + 0.277i·19-s + (−0.499 − 0.190i)21-s + (0.884 − 0.466i)23-s + 0.0472·25-s + (−0.888 + 0.458i)27-s − 0.141i·29-s − 1.07·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.119i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.119i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.992 + 0.119i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ -0.992 + 0.119i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0396323 - 0.658722i\)
\(L(\frac12)\) \(\approx\) \(0.0396323 - 0.658722i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.618 + 1.61i)T \)
23 \( 1 + (-4.24 + 2.23i)T \)
good5 \( 1 + 2.28T + 5T^{2} \)
7 \( 1 + 1.41iT - 7T^{2} \)
11 \( 1 + 1.95T + 11T^{2} \)
13 \( 1 - 1.23T + 13T^{2} \)
17 \( 1 + 7.73T + 17T^{2} \)
19 \( 1 - 1.20iT - 19T^{2} \)
29 \( 1 + 0.763iT - 29T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 + 7.61iT - 37T^{2} \)
41 \( 1 + 8.47iT - 41T^{2} \)
43 \( 1 - 5.78iT - 43T^{2} \)
47 \( 1 - 1.70iT - 47T^{2} \)
53 \( 1 - 2.28T + 53T^{2} \)
59 \( 1 - 8iT - 59T^{2} \)
61 \( 1 + 8.27iT - 61T^{2} \)
67 \( 1 - 3.36iT - 67T^{2} \)
71 \( 1 + 10.1iT - 71T^{2} \)
73 \( 1 + 2.76T + 73T^{2} \)
79 \( 1 + 12.3iT - 79T^{2} \)
83 \( 1 - 4.78T + 83T^{2} \)
89 \( 1 - 5.99T + 89T^{2} \)
97 \( 1 + 16.5iT - 97T^{2} \)
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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

−10.71949341779939755510912416942, −9.137680144335187407896411808898, −8.514241851454541095237697265579, −7.51740276237415005965373220423, −7.06786215614261090207326198864, −5.93833384520098556006259518591, −4.47818695718296167966229949562, −3.48562584613584815460545328866, −2.15564736879325964523126007044, −0.34409592766589972110314568277, 2.45928763435267401814922324179, 3.57881652777935217480998624550, 4.52927282664644909688462798321, 5.39984551895401539012659084779, 6.76646764798280456098595455778, 7.87969102783370133678787912593, 8.678553363963583699406186370680, 9.258608051203219799434330497524, 10.44406301325556244062951610287, 11.21250815681661669833432854485

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