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Properties

Label 2-1100-55.43-c1-0-17
Degree $2$
Conductor $1100$
Sign $-0.850 + 0.525i$
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  + (1 + i)3-s + (−3.31 − 3.31i)7-s − i·9-s + 3.31i·11-s + (−3.31 + 3.31i)13-s + (−3.31 − 3.31i)17-s − 6.63i·21-s + (−3 − 3i)23-s + (4 − 4i)27-s − 6.63·29-s − 4·31-s + (−3.31 + 3.31i)33-s + (−5 + 5i)37-s − 6.63·39-s + (3.31 − 3.31i)43-s + ⋯
L(s)  = 1  + (0.577 + 0.577i)3-s + (−1.25 − 1.25i)7-s − 0.333i·9-s + 1.00i·11-s + (−0.919 + 0.919i)13-s + (−0.804 − 0.804i)17-s − 1.44i·21-s + (−0.625 − 0.625i)23-s + (0.769 − 0.769i)27-s − 1.23·29-s − 0.718·31-s + (−0.577 + 0.577i)33-s + (−0.821 + 0.821i)37-s − 1.06·39-s + (0.505 − 0.505i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign: $-0.850 + 0.525i$
Analytic conductor: \(8.78354\)
Root analytic conductor: \(2.96370\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1100} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1100,\ (\ :1/2),\ -0.850 + 0.525i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3034178795\)
\(L(\frac12)\) \(\approx\) \(0.3034178795\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.31iT \)
good3 \( 1 + (-1 - i)T + 3iT^{2} \)
7 \( 1 + (3.31 + 3.31i)T + 7iT^{2} \)
13 \( 1 + (3.31 - 3.31i)T - 13iT^{2} \)
17 \( 1 + (3.31 + 3.31i)T + 17iT^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + (3 + 3i)T + 23iT^{2} \)
29 \( 1 + 6.63T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (5 - 5i)T - 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (-3.31 + 3.31i)T - 43iT^{2} \)
47 \( 1 + (5 - 5i)T - 47iT^{2} \)
53 \( 1 + (-3 - 3i)T + 53iT^{2} \)
59 \( 1 + 10iT - 59T^{2} \)
61 \( 1 + 13.2iT - 61T^{2} \)
67 \( 1 + (-3 + 3i)T - 67iT^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 + (3.31 - 3.31i)T - 73iT^{2} \)
79 \( 1 - 13.2T + 79T^{2} \)
83 \( 1 + (-3.31 + 3.31i)T - 83iT^{2} \)
89 \( 1 + 12iT - 89T^{2} \)
97 \( 1 + (5 - 5i)T - 97iT^{2} \)
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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

−9.511111937310037157547435763611, −9.128153186201444690152978080415, −7.73266036611995303004294112224, −6.92443483625134747948605020464, −6.49954160964974157061061271287, −4.84436734129907640698705987305, −4.12896102099316064028493988756, −3.37193149075343909793007263338, −2.16130833678251663393415989637, −0.11188106919991573900149237059, 2.03568262591685441099556083814, 2.82585600300257893474795111264, 3.71296577050104981333802268548, 5.41549153723501915197444201926, 5.85881157335167865238368594188, 6.93097659270782873510646729729, 7.78046252361795955343124638745, 8.632536573386262589730404150676, 9.149485635398973082813016033268, 10.09244750150881610184884238623

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