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Properties

Label 2-552-24.11-c1-0-7
Degree $2$
Conductor $552$
Sign $0.992 - 0.121i$
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  + (0.484 − 1.32i)2-s + (−1.59 − 0.676i)3-s + (−1.53 − 1.28i)4-s − 1.67·5-s + (−1.67 + 1.79i)6-s + 2.28i·7-s + (−2.45 + 1.41i)8-s + (2.08 + 2.15i)9-s + (−0.809 + 2.22i)10-s + 2.91i·11-s + (1.57 + 3.08i)12-s − 4.63i·13-s + (3.03 + 1.10i)14-s + (2.66 + 1.13i)15-s + (0.689 + 3.94i)16-s + 3.10i·17-s + ⋯
L(s)  = 1  + (0.342 − 0.939i)2-s + (−0.920 − 0.390i)3-s + (−0.765 − 0.643i)4-s − 0.747·5-s + (−0.681 + 0.731i)6-s + 0.864i·7-s + (−0.866 + 0.499i)8-s + (0.695 + 0.718i)9-s + (−0.256 + 0.702i)10-s + 0.877i·11-s + (0.453 + 0.891i)12-s − 1.28i·13-s + (0.812 + 0.295i)14-s + (0.688 + 0.291i)15-s + (0.172 + 0.985i)16-s + 0.752i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.121i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.121i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.992 - 0.121i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.992 - 0.121i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.732887 + 0.0446345i\)
\(L(\frac12)\) \(\approx\) \(0.732887 + 0.0446345i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.484 + 1.32i)T \)
3 \( 1 + (1.59 + 0.676i)T \)
23 \( 1 - T \)
good5 \( 1 + 1.67T + 5T^{2} \)
7 \( 1 - 2.28iT - 7T^{2} \)
11 \( 1 - 2.91iT - 11T^{2} \)
13 \( 1 + 4.63iT - 13T^{2} \)
17 \( 1 - 3.10iT - 17T^{2} \)
19 \( 1 - 7.70T + 19T^{2} \)
29 \( 1 + 8.53T + 29T^{2} \)
31 \( 1 - 7.88iT - 31T^{2} \)
37 \( 1 - 3.97iT - 37T^{2} \)
41 \( 1 - 5.01iT - 41T^{2} \)
43 \( 1 - 7.62T + 43T^{2} \)
47 \( 1 - 4.61T + 47T^{2} \)
53 \( 1 + 0.427T + 53T^{2} \)
59 \( 1 - 9.10iT - 59T^{2} \)
61 \( 1 - 9.06iT - 61T^{2} \)
67 \( 1 + 8.29T + 67T^{2} \)
71 \( 1 + 4.33T + 71T^{2} \)
73 \( 1 - 6.91T + 73T^{2} \)
79 \( 1 - 6.46iT - 79T^{2} \)
83 \( 1 - 12.7iT - 83T^{2} \)
89 \( 1 + 4.52iT - 89T^{2} \)
97 \( 1 + 8.58T + 97T^{2} \)
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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

−10.98872418498960531023986167056, −10.21504646437241498460831070009, −9.319270224613173573025306820848, −8.091195977322976158283435492631, −7.24044831548076028490438070852, −5.74002107678035965895952658132, −5.31027513548235345045152745761, −4.10835859066866082696238862678, −2.84007375790128794710268767918, −1.33258778903445573124066252940, 0.48324800916520260677936381274, 3.61181620918215539459546452959, 4.16232774350458879402837045075, 5.26249674961331096242747634039, 6.11287115325545353707149367944, 7.31823328846886882503958860616, 7.53156802443720069184251663669, 9.118255818443150139090837318261, 9.643700410291987501201429126111, 11.10240509890980230939095460085

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