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Properties

Label 2-1155-77.76-c1-0-38
Degree $2$
Conductor $1155$
Sign $0.903 - 0.428i$
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  − 0.370i·2-s + i·3-s + 1.86·4-s + i·5-s + 0.370·6-s + (2.39 + 1.12i)7-s − 1.43i·8-s − 9-s + 0.370·10-s + (2.11 − 2.55i)11-s + 1.86i·12-s − 5.18·13-s + (0.415 − 0.888i)14-s − 15-s + 3.19·16-s + 2.13·17-s + ⋯
L(s)  = 1  − 0.262i·2-s + 0.577i·3-s + 0.931·4-s + 0.447i·5-s + 0.151·6-s + (0.905 + 0.424i)7-s − 0.506i·8-s − 0.333·9-s + 0.117·10-s + (0.636 − 0.771i)11-s + 0.537i·12-s − 1.43·13-s + (0.111 − 0.237i)14-s − 0.258·15-s + 0.798·16-s + 0.517·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 - 0.428i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.903 - 0.428i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.903 - 0.428i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ 0.903 - 0.428i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.336172154\)
\(L(\frac12)\) \(\approx\) \(2.336172154\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
5 \( 1 - iT \)
7 \( 1 + (-2.39 - 1.12i)T \)
11 \( 1 + (-2.11 + 2.55i)T \)
good2 \( 1 + 0.370iT - 2T^{2} \)
13 \( 1 + 5.18T + 13T^{2} \)
17 \( 1 - 2.13T + 17T^{2} \)
19 \( 1 - 5.11T + 19T^{2} \)
23 \( 1 - 7.91T + 23T^{2} \)
29 \( 1 + 7.81iT - 29T^{2} \)
31 \( 1 - 7.84iT - 31T^{2} \)
37 \( 1 + 8.48T + 37T^{2} \)
41 \( 1 + 2.17T + 41T^{2} \)
43 \( 1 - 7.76iT - 43T^{2} \)
47 \( 1 - 5.62iT - 47T^{2} \)
53 \( 1 + 10.7T + 53T^{2} \)
59 \( 1 + 2.14iT - 59T^{2} \)
61 \( 1 + 8.26T + 61T^{2} \)
67 \( 1 + 2.80T + 67T^{2} \)
71 \( 1 - 9.32T + 71T^{2} \)
73 \( 1 - 10.0T + 73T^{2} \)
79 \( 1 - 8.98iT - 79T^{2} \)
83 \( 1 + 3.96T + 83T^{2} \)
89 \( 1 + 8.45iT - 89T^{2} \)
97 \( 1 - 8.31iT - 97T^{2} \)
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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

−9.888825967751309234798593796499, −9.262198277919302283648326430447, −8.135201728436926387943517036323, −7.39203674914537456116116777380, −6.55209771599573143757307275909, −5.50122886506470496623959059946, −4.78021521468329074180750549510, −3.33651131598566359887105989628, −2.74869624832430837502052006650, −1.39015792440269710044886346531, 1.22954754565115877821315439471, 2.11697320194248196958534128454, 3.37982765655300249654806666182, 4.93942857023753332051807394861, 5.31855186754072623369829217620, 6.73222005245062432818861982630, 7.32234077055353185060296420233, 7.70082442507467318011052963121, 8.811627990347595288447898588132, 9.703798685037530594215373973070

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