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Properties

Label 2-1107-123.122-c0-0-1
Degree $2$
Conductor $1107$
Sign $-1$
Analytic cond. $0.552464$
Root an. cond. $0.743279$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·2-s − 1.00·4-s + 1.41i·5-s + 1.41i·7-s − 2.00·10-s + 11-s − 1.41i·13-s − 2.00·14-s − 0.999·16-s − 1.41i·19-s − 1.41i·20-s + 1.41i·22-s − 1.00·25-s + 2.00·26-s − 1.41i·28-s + 29-s + ⋯
L(s)  = 1  + 1.41i·2-s − 1.00·4-s + 1.41i·5-s + 1.41i·7-s − 2.00·10-s + 11-s − 1.41i·13-s − 2.00·14-s − 0.999·16-s − 1.41i·19-s − 1.41i·20-s + 1.41i·22-s − 1.00·25-s + 2.00·26-s − 1.41i·28-s + 29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1107 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1107 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1107\)    =    \(3^{3} \cdot 41\)
Sign: $-1$
Analytic conductor: \(0.552464\)
Root analytic conductor: \(0.743279\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1107} (1106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1107,\ (\ :0),\ -1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.066916424\)
\(L(\frac12)\) \(\approx\) \(1.066916424\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
41 \( 1 + T \)
good2 \( 1 - 1.41iT - T^{2} \)
5 \( 1 - 1.41iT - T^{2} \)
7 \( 1 - 1.41iT - T^{2} \)
11 \( 1 - T + T^{2} \)
13 \( 1 + 1.41iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + 1.41iT - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T + T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 + T + T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + T + T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + 1.41iT - T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 - 1.41iT - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T + T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.35446548592590528618730264372, −9.364905114538635228932548891263, −8.565776419683803445246473209074, −7.915450636333524795051030047214, −6.77711623705930125261376907714, −6.54588564241700347792236004883, −5.63858256634926260625546451013, −4.84117083221838953839177045389, −3.21812261347978560846546441383, −2.45210827525021714618774031580, 1.11027879204958729344537448479, 1.71767846893019942380008488980, 3.50104886514408832170066061435, 4.24900000758527739037823096301, 4.68918558065948384903397131084, 6.29772276528224534033590768459, 7.14695446806623649430722994662, 8.369314183146887010125354279614, 9.046429113959001774016179977241, 9.852974903213684452926770078528

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