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Properties

Label 2-1155-5.4-c1-0-0
Degree $2$
Conductor $1155$
Sign $-0.0330 - 0.999i$
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  − 2.55i·2-s + i·3-s − 4.54·4-s + (0.0738 + 2.23i)5-s + 2.55·6-s + i·7-s + 6.50i·8-s − 9-s + (5.71 − 0.188i)10-s + 11-s − 4.54i·12-s − 3.07i·13-s + 2.55·14-s + (−2.23 + 0.0738i)15-s + 7.55·16-s − 2.96i·17-s + ⋯
L(s)  = 1  − 1.80i·2-s + 0.577i·3-s − 2.27·4-s + (0.0330 + 0.999i)5-s + 1.04·6-s + 0.377i·7-s + 2.29i·8-s − 0.333·9-s + (1.80 − 0.0597i)10-s + 0.301·11-s − 1.31i·12-s − 0.854i·13-s + 0.683·14-s + (−0.577 + 0.0190i)15-s + 1.88·16-s − 0.718i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0330 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0330 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.0330 - 0.999i$
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: \((2,\ 1155,\ (\ :1/2),\ -0.0330 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1962316700\)
\(L(\frac12)\) \(\approx\) \(0.1962316700\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0738 - 2.23i)T \)
7 \( 1 - iT \)
11 \( 1 - T \)
good2 \( 1 + 2.55iT - 2T^{2} \)
13 \( 1 + 3.07iT - 13T^{2} \)
17 \( 1 + 2.96iT - 17T^{2} \)
19 \( 1 + 6.66T + 19T^{2} \)
23 \( 1 - 3.16iT - 23T^{2} \)
29 \( 1 + 7.70T + 29T^{2} \)
31 \( 1 - 4.22T + 31T^{2} \)
37 \( 1 - 6.02iT - 37T^{2} \)
41 \( 1 + 9.67T + 41T^{2} \)
43 \( 1 - 2.22iT - 43T^{2} \)
47 \( 1 - 0.862iT - 47T^{2} \)
53 \( 1 + 7.20iT - 53T^{2} \)
59 \( 1 + 5.15T + 59T^{2} \)
61 \( 1 + 8.86T + 61T^{2} \)
67 \( 1 + 1.61iT - 67T^{2} \)
71 \( 1 - 1.95T + 71T^{2} \)
73 \( 1 - 3.60iT - 73T^{2} \)
79 \( 1 + 14.0T + 79T^{2} \)
83 \( 1 + 17.7iT - 83T^{2} \)
89 \( 1 + 7.48T + 89T^{2} \)
97 \( 1 - 19.2iT - 97T^{2} \)
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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

−10.20884919406193923980250686108, −9.575565734867185345502541444436, −8.787380896116090127777880521056, −7.87801619120023547416460279604, −6.52352524192087525325608632722, −5.43094318021696732937210769910, −4.45023673898370613679919340240, −3.47404799516253092551331781380, −2.86940388318373256583811584621, −1.84256227392076269503617803679, 0.083458161325640222589489348402, 1.73809145221643429039664848778, 4.01899901187301679620964132977, 4.54487464503440791419925783762, 5.65245399781823550880346198393, 6.32282985956808721953388392069, 7.02787240643981293647586964445, 7.86489785868609091112303437603, 8.640823693714668436375655766038, 8.974473235864597920957299234992

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