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Properties

Label 2-1100-11.4-c1-0-17
Degree $2$
Conductor $1100$
Sign $-0.987 + 0.156i$
Analytic cond. $8.78354$
Root an. cond. $2.96370$
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  + (0.954 − 2.93i)3-s + (−0.787 − 2.42i)7-s + (−5.29 − 3.85i)9-s + (2.91 + 1.57i)11-s + (−3.13 − 2.27i)13-s + (5.39 − 3.92i)17-s + (−0.918 + 2.82i)19-s − 7.87·21-s − 2.50·23-s + (−8.87 + 6.44i)27-s + (−1.00 − 3.08i)29-s + (−3.01 − 2.18i)31-s + (7.41 − 7.07i)33-s + (−0.630 − 1.94i)37-s + (−9.67 + 7.02i)39-s + ⋯
L(s)  = 1  + (0.551 − 1.69i)3-s + (−0.297 − 0.916i)7-s + (−1.76 − 1.28i)9-s + (0.880 + 0.474i)11-s + (−0.868 − 0.630i)13-s + (1.30 − 0.951i)17-s + (−0.210 + 0.648i)19-s − 1.71·21-s − 0.521·23-s + (−1.70 + 1.24i)27-s + (−0.185 − 0.571i)29-s + (−0.541 − 0.393i)31-s + (1.29 − 1.23i)33-s + (−0.103 − 0.319i)37-s + (−1.54 + 1.12i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.156i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 + 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign: $-0.987 + 0.156i$
Analytic conductor: \(8.78354\)
Root analytic conductor: \(2.96370\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1100} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1100,\ (\ :1/2),\ -0.987 + 0.156i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.587130057\)
\(L(\frac12)\) \(\approx\) \(1.587130057\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
11 \( 1 + (-2.91 - 1.57i)T \)
good3 \( 1 + (-0.954 + 2.93i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + (0.787 + 2.42i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (3.13 + 2.27i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-5.39 + 3.92i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.918 - 2.82i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 2.50T + 23T^{2} \)
29 \( 1 + (1.00 + 3.08i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (3.01 + 2.18i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.630 + 1.94i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (3.15 - 9.70i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 6.30T + 43T^{2} \)
47 \( 1 + (0.894 - 2.75i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-7.36 - 5.35i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.44 + 7.52i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-5.03 + 3.65i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 7.40T + 67T^{2} \)
71 \( 1 + (-5.48 + 3.98i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-4.28 - 13.1i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-2.17 - 1.57i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-2.08 + 1.51i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 2.81T + 89T^{2} \)
97 \( 1 + (5.31 + 3.85i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.574703445641296651314989779888, −8.265189751690257269964055731632, −7.70030053797639874606434592799, −7.10173174228256397135697120567, −6.42787007125683610233051832749, −5.37184577149157435594411052852, −3.89192705115544463894265134191, −2.92048416757130542480752107005, −1.75573697656360834220703187259, −0.64921350695099055962329605544, 2.17491140341382675264651152013, 3.35246615918605315756715644897, 3.92911803529490097884212909734, 5.06637385098441315245197436912, 5.68726348424750505210417909062, 6.84399854214729492952180096294, 8.189842135971418573514465817143, 8.878840203466239108361000426536, 9.357552634994354402188624761357, 10.11152427532529935858780139844

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