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Properties

Label 2-1155-385.384-c1-0-65
Degree $2$
Conductor $1155$
Sign $0.996 - 0.0881i$
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  + 1.17·2-s + 3-s − 0.623·4-s + (2.20 + 0.344i)5-s + 1.17·6-s + (1.71 + 2.01i)7-s − 3.07·8-s + 9-s + (2.59 + 0.404i)10-s + (1.48 − 2.96i)11-s − 0.623·12-s − 3.43i·13-s + (2.01 + 2.36i)14-s + (2.20 + 0.344i)15-s − 2.36·16-s − 3.10i·17-s + ⋯
L(s)  = 1  + 0.829·2-s + 0.577·3-s − 0.311·4-s + (0.988 + 0.154i)5-s + 0.478·6-s + (0.649 + 0.760i)7-s − 1.08·8-s + 0.333·9-s + (0.819 + 0.127i)10-s + (0.446 − 0.894i)11-s − 0.180·12-s − 0.952i·13-s + (0.538 + 0.631i)14-s + (0.570 + 0.0890i)15-s − 0.590·16-s − 0.752i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.996 - 0.0881i$
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: \((2,\ 1155,\ (\ :1/2),\ 0.996 - 0.0881i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.346796210\)
\(L(\frac12)\) \(\approx\) \(3.346796210\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.20 - 0.344i)T \)
7 \( 1 + (-1.71 - 2.01i)T \)
11 \( 1 + (-1.48 + 2.96i)T \)
good2 \( 1 - 1.17T + 2T^{2} \)
13 \( 1 + 3.43iT - 13T^{2} \)
17 \( 1 + 3.10iT - 17T^{2} \)
19 \( 1 - 5.32T + 19T^{2} \)
23 \( 1 - 5.88iT - 23T^{2} \)
29 \( 1 - 5.81iT - 29T^{2} \)
31 \( 1 - 0.722iT - 31T^{2} \)
37 \( 1 - 1.11iT - 37T^{2} \)
41 \( 1 + 3.34T + 41T^{2} \)
43 \( 1 + 7.30T + 43T^{2} \)
47 \( 1 - 6.60T + 47T^{2} \)
53 \( 1 - 11.7iT - 53T^{2} \)
59 \( 1 + 14.7iT - 59T^{2} \)
61 \( 1 + 5.25T + 61T^{2} \)
67 \( 1 - 1.98iT - 67T^{2} \)
71 \( 1 + 14.9T + 71T^{2} \)
73 \( 1 + 7.27iT - 73T^{2} \)
79 \( 1 + 11.5iT - 79T^{2} \)
83 \( 1 + 7.39iT - 83T^{2} \)
89 \( 1 - 3.65iT - 89T^{2} \)
97 \( 1 + 3.77T + 97T^{2} \)
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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

−9.508747258016611172914781023230, −9.112885902975297331020168284077, −8.336531120454019378414603798071, −7.30646473316548891634376751717, −6.06010018045081001039694119381, −5.44352271060385485953627117574, −4.86728570418096682313557559128, −3.32716128945618316408246863698, −2.92749109989922130662121964719, −1.40801338213313037891588564411, 1.41797271855279845128746928317, 2.51651202948052746011504069553, 3.90677510475794976995102851223, 4.46605521036112060965331935247, 5.30064117841618133412844638078, 6.37727440168282419163091898505, 7.12721830770984590823573738571, 8.278779320365709904562657015498, 9.018510566191040947341396370329, 9.774621733793974785027517677499

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