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Properties

Label 2-1155-5.4-c1-0-35
Degree $2$
Conductor $1155$
Sign $0.823 + 0.566i$
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  − 0.0342i·2-s + i·3-s + 1.99·4-s + (−1.84 − 1.26i)5-s + 0.0342·6-s + i·7-s − 0.137i·8-s − 9-s + (−0.0434 + 0.0631i)10-s + 11-s + 1.99i·12-s − 5.23i·13-s + 0.0342·14-s + (1.26 − 1.84i)15-s + 3.99·16-s − 5.86i·17-s + ⋯
L(s)  = 1  − 0.0242i·2-s + 0.577i·3-s + 0.999·4-s + (−0.823 − 0.566i)5-s + 0.0140·6-s + 0.377i·7-s − 0.0484i·8-s − 0.333·9-s + (−0.0137 + 0.0199i)10-s + 0.301·11-s + 0.577i·12-s − 1.45i·13-s + 0.00916·14-s + (0.327 − 0.475i)15-s + 0.998·16-s − 1.42i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 + 0.566i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.823 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.823 + 0.566i$
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: \((2,\ 1155,\ (\ :1/2),\ 0.823 + 0.566i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.730090796\)
\(L(\frac12)\) \(\approx\) \(1.730090796\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.84 + 1.26i)T \)
7 \( 1 - iT \)
11 \( 1 - T \)
good2 \( 1 + 0.0342iT - 2T^{2} \)
13 \( 1 + 5.23iT - 13T^{2} \)
17 \( 1 + 5.86iT - 17T^{2} \)
19 \( 1 + 1.91T + 19T^{2} \)
23 \( 1 + 2.87iT - 23T^{2} \)
29 \( 1 - 6.16T + 29T^{2} \)
31 \( 1 - 8.30T + 31T^{2} \)
37 \( 1 - 8.61iT - 37T^{2} \)
41 \( 1 + 1.69T + 41T^{2} \)
43 \( 1 + 6.55iT - 43T^{2} \)
47 \( 1 + 6.80iT - 47T^{2} \)
53 \( 1 + 2.02iT - 53T^{2} \)
59 \( 1 + 6.11T + 59T^{2} \)
61 \( 1 - 12.1T + 61T^{2} \)
67 \( 1 + 0.347iT - 67T^{2} \)
71 \( 1 + 15.6T + 71T^{2} \)
73 \( 1 - 12.7iT - 73T^{2} \)
79 \( 1 - 1.95T + 79T^{2} \)
83 \( 1 + 3.09iT - 83T^{2} \)
89 \( 1 - 9.12T + 89T^{2} \)
97 \( 1 + 10.8iT - 97T^{2} \)
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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

−9.971358343304651632301927796829, −8.663462322513939864078641855841, −8.238356748580889131886009908604, −7.26919774037841755049442814925, −6.38983073744877436798852300253, −5.33028689457997812888671357043, −4.60655103695328767294157674795, −3.32978409083473839138889901401, −2.62970089254369689972255737422, −0.803213860714155043350753274536, 1.39226429772656913916339612913, 2.49213274561495377533640244625, 3.62216617351708897300364136961, 4.48680010029787923571996589175, 6.27542735709173867575826698545, 6.43180230397027147424692830930, 7.37140803989549872124840772573, 7.973390112647798705514850484675, 8.851404040392954514813756179082, 10.12650128142152810892361427582

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