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Properties

Label 2-1104-12.11-c1-0-11
Degree $2$
Conductor $1104$
Sign $-0.288 - 0.957i$
Analytic cond. $8.81548$
Root an. cond. $2.96908$
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  + (1.68 + 0.396i)3-s + 3.46i·7-s + (2.68 + 1.33i)9-s − 4·11-s − 5.37·13-s + 1.58i·17-s + 6.63i·19-s + (−1.37 + 5.84i)21-s − 23-s + 5·25-s + (4 + 3.31i)27-s + 5.84i·29-s − 0.792i·31-s + (−6.74 − 1.58i)33-s − 4.74·37-s + ⋯
L(s)  = 1  + (0.973 + 0.228i)3-s + 1.30i·7-s + (0.895 + 0.445i)9-s − 1.20·11-s − 1.49·13-s + 0.384i·17-s + 1.52i·19-s + (−0.299 + 1.27i)21-s − 0.208·23-s + 25-s + (0.769 + 0.638i)27-s + 1.08i·29-s − 0.142i·31-s + (−1.17 − 0.275i)33-s − 0.780·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.288 - 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.288 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1104\)    =    \(2^{4} \cdot 3 \cdot 23\)
Sign: $-0.288 - 0.957i$
Analytic conductor: \(8.81548\)
Root analytic conductor: \(2.96908\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1104} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1104,\ (\ :1/2),\ -0.288 - 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.743174801\)
\(L(\frac12)\) \(\approx\) \(1.743174801\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.68 - 0.396i)T \)
23 \( 1 + T \)
good5 \( 1 - 5T^{2} \)
7 \( 1 - 3.46iT - 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + 5.37T + 13T^{2} \)
17 \( 1 - 1.58iT - 17T^{2} \)
19 \( 1 - 6.63iT - 19T^{2} \)
29 \( 1 - 5.84iT - 29T^{2} \)
31 \( 1 + 0.792iT - 31T^{2} \)
37 \( 1 + 4.74T + 37T^{2} \)
41 \( 1 + 11.1iT - 41T^{2} \)
43 \( 1 + 1.87iT - 43T^{2} \)
47 \( 1 - 10.1T + 47T^{2} \)
53 \( 1 + 8.51iT - 53T^{2} \)
59 \( 1 - 13.4T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 - 15.1iT - 67T^{2} \)
71 \( 1 + 2.11T + 71T^{2} \)
73 \( 1 + 1.37T + 73T^{2} \)
79 \( 1 - 9.80iT - 79T^{2} \)
83 \( 1 - 10.7T + 83T^{2} \)
89 \( 1 + 13.2iT - 89T^{2} \)
97 \( 1 - 7.48T + 97T^{2} \)
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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

−10.24854690173430015575238357310, −9.078125246130699775677227303796, −8.580100595807665264996599428006, −7.75067939788798475188237860690, −7.01091037791130124301123589013, −5.55041769820569216752184717865, −5.08124531446502471159202627107, −3.76052889938682801995349893489, −2.66439628617244599102531188781, −2.05419156038094758139734062728, 0.64477207867927264347941239780, 2.36722822307091642773417915504, 3.07265720896718235790406933836, 4.41476397068093875732836342005, 4.96022757697945554921422072076, 6.58439965815713154683139342253, 7.45325936636833501956990917100, 7.64024809356573651177290568134, 8.781814653573444094735700100443, 9.656496242113180133441665010608

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