Properties

Label 2-1155-5.4-c1-0-51
Degree $2$
Conductor $1155$
Sign $-0.920 + 0.391i$
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  − 1.28i·2-s i·3-s + 0.359·4-s + (2.05 − 0.876i)5-s − 1.28·6-s i·7-s − 3.02i·8-s − 9-s + (−1.12 − 2.63i)10-s + 11-s − 0.359i·12-s − 4.33i·13-s − 1.28·14-s + (−0.876 − 2.05i)15-s − 3.15·16-s + 6.42i·17-s + ⋯
L(s)  = 1  − 0.905i·2-s − 0.577i·3-s + 0.179·4-s + (0.920 − 0.391i)5-s − 0.522·6-s − 0.377i·7-s − 1.06i·8-s − 0.333·9-s + (−0.354 − 0.833i)10-s + 0.301·11-s − 0.103i·12-s − 1.20i·13-s − 0.342·14-s + (−0.226 − 0.531i)15-s − 0.788·16-s + 1.55i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 + 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.920 + 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.920 + 0.391i$
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: \((2,\ 1155,\ (\ :1/2),\ -0.920 + 0.391i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.125285971\)
\(L(\frac12)\) \(\approx\) \(2.125285971\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.05 + 0.876i)T \)
7 \( 1 + iT \)
11 \( 1 - T \)
good2 \( 1 + 1.28iT - 2T^{2} \)
13 \( 1 + 4.33iT - 13T^{2} \)
17 \( 1 - 6.42iT - 17T^{2} \)
19 \( 1 + 2.73T + 19T^{2} \)
23 \( 1 + 3.91iT - 23T^{2} \)
29 \( 1 + 4.11T + 29T^{2} \)
31 \( 1 - 8.94T + 31T^{2} \)
37 \( 1 - 11.0iT - 37T^{2} \)
41 \( 1 + 3.74T + 41T^{2} \)
43 \( 1 - 10.5iT - 43T^{2} \)
47 \( 1 + 11.8iT - 47T^{2} \)
53 \( 1 + 8.18iT - 53T^{2} \)
59 \( 1 - 4.77T + 59T^{2} \)
61 \( 1 - 7.12T + 61T^{2} \)
67 \( 1 + 5.84iT - 67T^{2} \)
71 \( 1 + 0.841T + 71T^{2} \)
73 \( 1 + 0.456iT - 73T^{2} \)
79 \( 1 - 6.39T + 79T^{2} \)
83 \( 1 - 9.87iT - 83T^{2} \)
89 \( 1 + 8.67T + 89T^{2} \)
97 \( 1 - 2.64iT - 97T^{2} \)
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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

−9.876876636249059066896897063458, −8.538064029515617345596462252569, −8.040363185500184260203467608602, −6.54624678569972600356573729349, −6.39175928948011052112324292932, −5.13925605526166867749490133400, −3.91277377890053295919327606385, −2.82459587855640765935591044649, −1.85646011495796016919653965882, −0.926116144798390161167848741346, 1.97108856321362031002736671427, 2.84628019105511542843250963278, 4.31631900590688207994269160903, 5.33577067395640060226334428247, 5.94882734694534761571507988006, 6.82468373263804928789114098665, 7.37787488261746491031226228617, 8.670526645088723377366928528982, 9.242235013545378725273271331378, 9.915953305244479486878147944888

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