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Properties

Label 2-1100-55.9-c1-0-17
Degree $2$
Conductor $1100$
Sign $-0.908 + 0.418i$
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  + (−0.224 − 0.309i)3-s + (1.67 − 2.30i)7-s + (0.881 − 2.71i)9-s + (−3.23 + 0.726i)11-s + (−4.39 − 1.42i)13-s + (−4.39 + 1.42i)17-s + (−2.30 + 1.67i)19-s − 1.09·21-s − 6.47i·23-s + (−2.12 + 0.690i)27-s + (5.16 + 3.75i)29-s + (−1.80 + 5.56i)31-s + (0.951 + 0.836i)33-s + (2.85 − 3.92i)37-s + (0.545 + 1.67i)39-s + ⋯
L(s)  = 1  + (−0.129 − 0.178i)3-s + (0.634 − 0.872i)7-s + (0.293 − 0.904i)9-s + (−0.975 + 0.219i)11-s + (−1.21 − 0.395i)13-s + (−1.06 + 0.346i)17-s + (−0.529 + 0.384i)19-s − 0.237·21-s − 1.34i·23-s + (−0.409 + 0.132i)27-s + (0.958 + 0.696i)29-s + (−0.324 + 0.999i)31-s + (0.165 + 0.145i)33-s + (0.469 − 0.645i)37-s + (0.0872 + 0.268i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.908 + 0.418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign: $-0.908 + 0.418i$
Analytic conductor: \(8.78354\)
Root analytic conductor: \(2.96370\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1100} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1100,\ (\ :1/2),\ -0.908 + 0.418i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7244942963\)
\(L(\frac12)\) \(\approx\) \(0.7244942963\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.23 - 0.726i)T \)
good3 \( 1 + (0.224 + 0.309i)T + (-0.927 + 2.85i)T^{2} \)
7 \( 1 + (-1.67 + 2.30i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (4.39 + 1.42i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (4.39 - 1.42i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (2.30 - 1.67i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 6.47iT - 23T^{2} \)
29 \( 1 + (-5.16 - 3.75i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.80 - 5.56i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-2.85 + 3.92i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (5.16 - 3.75i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + (2.12 + 2.92i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (6.74 + 2.19i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (8.16 + 5.93i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-1.42 - 4.39i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 4.94iT - 67T^{2} \)
71 \( 1 + (2.66 + 8.19i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-7.10 + 9.78i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-4.28 + 13.1i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (15.2 - 4.95i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 - 8.47T + 89T^{2} \)
97 \( 1 + (-5.29 - 1.71i)T + (78.4 + 57.0i)T^{2} \)
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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

−9.597205011033792601183306951051, −8.565002237551191646927108553744, −7.80394493709062522645858948610, −6.98953992465266863211235761576, −6.31266137982785430723426080862, −4.92499439220801554896808512946, −4.45598452499167117688421750496, −3.12742296420178603300386141437, −1.86803300705302400776143477944, −0.30042239636346177958751512585, 2.03858344834147726246297022266, 2.67079674485366325858255765023, 4.42327543414284848216528873210, 4.99112385847748536340746477806, 5.76450122077084477193815441607, 6.99660523278297221714829990435, 7.81303023152984408120465151388, 8.477239460117511988290287045104, 9.474632539635582677820325547112, 10.14160630248436462475799703112

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