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Properties

Label 2-1155-385.384-c1-0-22
Degree $2$
Conductor $1155$
Sign $-0.255 - 0.966i$
Analytic cond. $9.22272$
Root an. cond. $3.03689$
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  + 2.43·2-s − 3-s + 3.90·4-s + (−2.21 − 0.290i)5-s − 2.43·6-s + (−1.43 + 2.22i)7-s + 4.64·8-s + 9-s + (−5.38 − 0.705i)10-s + (−1.89 + 2.72i)11-s − 3.90·12-s + 2.23i·13-s + (−3.48 + 5.40i)14-s + (2.21 + 0.290i)15-s + 3.46·16-s + 4.51i·17-s + ⋯
L(s)  = 1  + 1.71·2-s − 0.577·3-s + 1.95·4-s + (−0.991 − 0.129i)5-s − 0.992·6-s + (−0.542 + 0.840i)7-s + 1.64·8-s + 0.333·9-s + (−1.70 − 0.223i)10-s + (−0.571 + 0.820i)11-s − 1.12·12-s + 0.619i·13-s + (−0.932 + 1.44i)14-s + (0.572 + 0.0749i)15-s + 0.866·16-s + 1.09i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.255 - 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.255 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.255 - 0.966i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ -0.255 - 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.170725877\)
\(L(\frac12)\) \(\approx\) \(2.170725877\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 + (2.21 + 0.290i)T \)
7 \( 1 + (1.43 - 2.22i)T \)
11 \( 1 + (1.89 - 2.72i)T \)
good2 \( 1 - 2.43T + 2T^{2} \)
13 \( 1 - 2.23iT - 13T^{2} \)
17 \( 1 - 4.51iT - 17T^{2} \)
19 \( 1 - 0.900T + 19T^{2} \)
23 \( 1 - 2.09iT - 23T^{2} \)
29 \( 1 - 3.99iT - 29T^{2} \)
31 \( 1 + 8.71iT - 31T^{2} \)
37 \( 1 - 7.32iT - 37T^{2} \)
41 \( 1 + 0.222T + 41T^{2} \)
43 \( 1 + 4.66T + 43T^{2} \)
47 \( 1 - 6.60T + 47T^{2} \)
53 \( 1 + 3.38iT - 53T^{2} \)
59 \( 1 - 8.40iT - 59T^{2} \)
61 \( 1 - 2.18T + 61T^{2} \)
67 \( 1 + 11.5iT - 67T^{2} \)
71 \( 1 - 2.71T + 71T^{2} \)
73 \( 1 + 4.36iT - 73T^{2} \)
79 \( 1 + 4.44iT - 79T^{2} \)
83 \( 1 - 7.79iT - 83T^{2} \)
89 \( 1 - 1.90iT - 89T^{2} \)
97 \( 1 + 12.7T + 97T^{2} \)
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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

−10.36713390994981712881674852723, −9.292129455349179249323158683224, −8.117180982048067191478158675580, −7.16694465520264185691134073706, −6.44127077423939327653434948914, −5.62200705614802291751330102924, −4.83234868769812291454805935095, −4.08287694376546008143435457761, −3.20290840509850917926602668162, −2.01097930842340830465403296590, 0.54486904797197521463936552751, 2.81179042516816123085394762284, 3.48266841017128656837077188440, 4.35005197051407610086911781890, 5.15568923302932470567049721600, 5.95288070458827916504341383998, 6.96066230371392142948808946533, 7.36649263712963983763030005179, 8.488036485981582624158224381937, 9.952859352227733212968889338710

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