Properties

Label 2-1155-77.76-c1-0-7
Degree $2$
Conductor $1155$
Sign $0.999 - 0.0403i$
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  − 2.60i·2-s + i·3-s − 4.77·4-s i·5-s + 2.60·6-s + (−2.60 − 0.474i)7-s + 7.22i·8-s − 9-s − 2.60·10-s + (3.23 − 0.726i)11-s − 4.77i·12-s − 6.07·13-s + (−1.23 + 6.77i)14-s + 15-s + 9.24·16-s − 4.27·17-s + ⋯
L(s)  = 1  − 1.84i·2-s + 0.577i·3-s − 2.38·4-s − 0.447i·5-s + 1.06·6-s + (−0.983 − 0.179i)7-s + 2.55i·8-s − 0.333·9-s − 0.823·10-s + (0.975 − 0.219i)11-s − 1.37i·12-s − 1.68·13-s + (−0.330 + 1.81i)14-s + 0.258·15-s + 2.31·16-s − 1.03·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0403i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.999 - 0.0403i$
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: \((2,\ 1155,\ (\ :1/2),\ 0.999 - 0.0403i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4942073038\)
\(L(\frac12)\) \(\approx\) \(0.4942073038\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.60 + 0.474i)T \)
11 \( 1 + (-3.23 + 0.726i)T \)
good2 \( 1 + 2.60iT - 2T^{2} \)
13 \( 1 + 6.07T + 13T^{2} \)
17 \( 1 + 4.27T + 17T^{2} \)
19 \( 1 - 2.82T + 19T^{2} \)
23 \( 1 - 5.77T + 23T^{2} \)
29 \( 1 - 6.49iT - 29T^{2} \)
31 \( 1 - 2.66iT - 31T^{2} \)
37 \( 1 - 2.66T + 37T^{2} \)
41 \( 1 - 8.37T + 41T^{2} \)
43 \( 1 - 11.9iT - 43T^{2} \)
47 \( 1 - 12.6iT - 47T^{2} \)
53 \( 1 + 13.1T + 53T^{2} \)
59 \( 1 + 4.89iT - 59T^{2} \)
61 \( 1 + 7.86T + 61T^{2} \)
67 \( 1 - 2.92T + 67T^{2} \)
71 \( 1 + 11.1T + 71T^{2} \)
73 \( 1 + 4.53T + 73T^{2} \)
79 \( 1 + 7.42iT - 79T^{2} \)
83 \( 1 - 9.31T + 83T^{2} \)
89 \( 1 + 0.418iT - 89T^{2} \)
97 \( 1 + 2.43iT - 97T^{2} \)
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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

−9.636366518798445902020686756235, −9.415853698412273047624178637660, −8.893057086804726056933313523946, −7.50491829891143593937627979503, −6.26781692403086196654124543707, −4.85055848726916521341904335487, −4.49643501465602060857831025949, −3.30374221365016117182495848126, −2.72155408546757369413322081673, −1.20537364861600472949678927314, 0.24066639516278174519600788012, 2.55313041173856851520477604993, 3.94480202929249222087679059037, 4.95476827427470876677107799492, 5.93240478391794063457577966030, 6.63895593342753211527414052910, 7.15526358117494338135463251081, 7.69989926367120764442601687336, 8.963042247637569977454057253488, 9.330808031729991207522070158505

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