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Properties

Label 2-1104-1104.275-c0-0-4
Degree $2$
Conductor $1104$
Sign $-0.608 + 0.793i$
Analytic cond. $0.550967$
Root an. cond. $0.742272$
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  + (−0.866 − 0.5i)2-s + (0.5 − 0.866i)3-s + (0.499 + 0.866i)4-s + (−0.866 + 0.499i)6-s − 0.999i·8-s + (−0.499 − 0.866i)9-s + 0.999·12-s + (−1.36 − 1.36i)13-s + (−0.5 + 0.866i)16-s + 0.999i·18-s − i·23-s + (−0.866 − 0.5i)24-s + i·25-s + (0.499 + 1.86i)26-s − 0.999·27-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.5 − 0.866i)3-s + (0.499 + 0.866i)4-s + (−0.866 + 0.499i)6-s − 0.999i·8-s + (−0.499 − 0.866i)9-s + 0.999·12-s + (−1.36 − 1.36i)13-s + (−0.5 + 0.866i)16-s + 0.999i·18-s − i·23-s + (−0.866 − 0.5i)24-s + i·25-s + (0.499 + 1.86i)26-s − 0.999·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1104\)    =    \(2^{4} \cdot 3 \cdot 23\)
Sign: $-0.608 + 0.793i$
Analytic conductor: \(0.550967\)
Root analytic conductor: \(0.742272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1104} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1104,\ (\ :0),\ -0.608 + 0.793i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7145431917\)
\(L(\frac12)\) \(\approx\) \(0.7145431917\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + iT \)
good5 \( 1 - iT^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + iT^{2} \)
13 \( 1 + (1.36 + 1.36i)T + iT^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + iT^{2} \)
29 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
31 \( 1 + iT - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 - 1.73T + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + T + T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (1 - i)T - iT^{2} \)
61 \( 1 - iT^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - 1.73iT - T^{2} \)
73 \( 1 - iT - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - T^{2} \)
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

−9.738342944041083073591959039419, −8.924829194605546045027265157339, −7.977250319463199175503842189407, −7.64955638697596472854254751437, −6.73783632087172190304579041370, −5.74006792219693117981841162803, −4.24511233503845702034651190873, −2.88780712157090554480110336722, −2.39451296796351482119416629500, −0.802823269340315809142035729570, 1.90966467429491200510050020107, 3.00046122413154011339304285195, 4.50887176910348176885374666122, 5.09105351369225793878957679271, 6.32251789470612247528353732986, 7.20581715289107558811136432261, 7.967339132702009741878043590635, 8.937008196710433602971346481505, 9.349846556242287415692296971083, 10.13131749847430934743885723694

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