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Properties

Label 2-1155-1155.1154-c0-0-11
Degree $2$
Conductor $1155$
Sign $-0.866 - 0.5i$
Analytic cond. $0.576420$
Root an. cond. $0.759223$
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  − 1.73i·2-s − i·3-s − 1.99·4-s + (0.866 + 0.5i)5-s − 1.73·6-s − i·7-s + 1.73i·8-s − 9-s + (0.866 − 1.49i)10-s + 11-s + 1.99i·12-s − i·13-s − 1.73·14-s + (0.5 − 0.866i)15-s + 0.999·16-s + ⋯
L(s)  = 1  − 1.73i·2-s − i·3-s − 1.99·4-s + (0.866 + 0.5i)5-s − 1.73·6-s − i·7-s + 1.73i·8-s − 9-s + (0.866 − 1.49i)10-s + 11-s + 1.99i·12-s − i·13-s − 1.73·14-s + (0.5 − 0.866i)15-s + 0.999·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.866 - 0.5i$
Analytic conductor: \(0.576420\)
Root analytic conductor: \(0.759223\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (1154, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :0),\ -0.866 - 0.5i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.033070771\)
\(L(\frac12)\) \(\approx\) \(1.033070771\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + iT \)
11 \( 1 - T \)
good2 \( 1 + 1.73iT - T^{2} \)
13 \( 1 + iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + 1.73T + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 1.73iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - 1.73T + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + 1.73iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
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

−9.939598384940460824239817967461, −8.857755485875495821709834802233, −8.160427430607420577966108030901, −6.84908906968068297899502260078, −6.32099296482791001755868260594, −4.99174044644371470207611855904, −3.82067375058755751709208077572, −2.91370265807162717433781706912, −1.96201075347461263757079181143, −0.995465327582597890749757200938, 2.26491839036887251555052154813, 4.09896801006287131778281496356, 4.66475066800720105042113560230, 5.63142671380701312295180660042, 6.16889981728502958607626062766, 6.82629250369933062345625548732, 8.340150741283428704262259854052, 8.820808828244882075670875823415, 9.221911080790382031990196671248, 9.965912350350954083173225385917

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