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Properties

Label 2-1155-5.4-c1-0-40
Degree $2$
Conductor $1155$
Sign $-0.773 + 0.634i$
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  − 1.93i·2-s + i·3-s − 1.74·4-s + (1.72 − 1.41i)5-s + 1.93·6-s + i·7-s − 0.497i·8-s − 9-s + (−2.74 − 3.34i)10-s + 11-s − 1.74i·12-s + 0.486i·13-s + 1.93·14-s + (1.41 + 1.72i)15-s − 4.44·16-s − 4.43i·17-s + ⋯
L(s)  = 1  − 1.36i·2-s + 0.577i·3-s − 0.871·4-s + (0.773 − 0.634i)5-s + 0.789·6-s + 0.377i·7-s − 0.175i·8-s − 0.333·9-s + (−0.867 − 1.05i)10-s + 0.301·11-s − 0.503i·12-s + 0.134i·13-s + 0.517·14-s + (0.366 + 0.446i)15-s − 1.11·16-s − 1.07i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.773 + 0.634i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.773 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.773 + 0.634i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (694, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ -0.773 + 0.634i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.738533074\)
\(L(\frac12)\) \(\approx\) \(1.738533074\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 + 1.41i)T \)
7 \( 1 - iT \)
11 \( 1 - T \)
good2 \( 1 + 1.93iT - 2T^{2} \)
13 \( 1 - 0.486iT - 13T^{2} \)
17 \( 1 + 4.43iT - 17T^{2} \)
19 \( 1 + 3.34T + 19T^{2} \)
23 \( 1 + 5.94iT - 23T^{2} \)
29 \( 1 - 3.07T + 29T^{2} \)
31 \( 1 - 6.58T + 31T^{2} \)
37 \( 1 + 6.41iT - 37T^{2} \)
41 \( 1 - 8.83T + 41T^{2} \)
43 \( 1 - 5.13iT - 43T^{2} \)
47 \( 1 + 11.8iT - 47T^{2} \)
53 \( 1 - 6.16iT - 53T^{2} \)
59 \( 1 + 8.45T + 59T^{2} \)
61 \( 1 - 2.24T + 61T^{2} \)
67 \( 1 - 6.81iT - 67T^{2} \)
71 \( 1 - 3.62T + 71T^{2} \)
73 \( 1 + 15.7iT - 73T^{2} \)
79 \( 1 + 12.0T + 79T^{2} \)
83 \( 1 - 14.0iT - 83T^{2} \)
89 \( 1 - 3.85T + 89T^{2} \)
97 \( 1 - 3.75iT - 97T^{2} \)
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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

−9.563134517274494728854614352669, −9.089367678164104290784149646550, −8.328044266280812156547133390100, −6.80111324361335090762574667276, −5.92777494687213558314132615106, −4.76528304740650268383936172846, −4.22909664813893762208472941200, −2.85987399504316400491750028026, −2.18373924353273426948886183010, −0.77079174531822217145335029042, 1.60008050491121863540583723922, 2.85275898112188716116970732261, 4.27591629147276266721700339450, 5.45724507916713926685454000690, 6.25947696975030595355259597780, 6.59814582374333472491619384141, 7.53059974029892167123170456240, 8.158107130709003481001820785059, 9.038121059989744768827884948011, 9.980176913375350960766977737546

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