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Properties

Label 2-1155-385.384-c1-0-35
Degree $2$
Conductor $1155$
Sign $0.870 - 0.492i$
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  − 2.66·2-s + 3-s + 5.10·4-s + (−1.18 − 1.89i)5-s − 2.66·6-s + (2.09 + 1.61i)7-s − 8.26·8-s + 9-s + (3.15 + 5.05i)10-s + (1.90 + 2.71i)11-s + 5.10·12-s − 0.864i·13-s + (−5.59 − 4.29i)14-s + (−1.18 − 1.89i)15-s + 11.8·16-s − 1.53i·17-s + ⋯
L(s)  = 1  − 1.88·2-s + 0.577·3-s + 2.55·4-s + (−0.530 − 0.847i)5-s − 1.08·6-s + (0.793 + 0.608i)7-s − 2.92·8-s + 0.333·9-s + (0.998 + 1.59i)10-s + (0.573 + 0.819i)11-s + 1.47·12-s − 0.239i·13-s + (−1.49 − 1.14i)14-s + (−0.306 − 0.489i)15-s + 2.95·16-s − 0.372i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 - 0.492i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.870 - 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.870 - 0.492i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ 0.870 - 0.492i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8524807254\)
\(L(\frac12)\) \(\approx\) \(0.8524807254\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 + (1.18 + 1.89i)T \)
7 \( 1 + (-2.09 - 1.61i)T \)
11 \( 1 + (-1.90 - 2.71i)T \)
good2 \( 1 + 2.66T + 2T^{2} \)
13 \( 1 + 0.864iT - 13T^{2} \)
17 \( 1 + 1.53iT - 17T^{2} \)
19 \( 1 - 2.68T + 19T^{2} \)
23 \( 1 - 7.32iT - 23T^{2} \)
29 \( 1 - 1.42iT - 29T^{2} \)
31 \( 1 + 0.551iT - 31T^{2} \)
37 \( 1 - 5.97iT - 37T^{2} \)
41 \( 1 + 0.607T + 41T^{2} \)
43 \( 1 + 0.823T + 43T^{2} \)
47 \( 1 - 8.37T + 47T^{2} \)
53 \( 1 + 8.21iT - 53T^{2} \)
59 \( 1 - 14.3iT - 59T^{2} \)
61 \( 1 + 6.95T + 61T^{2} \)
67 \( 1 + 7.78iT - 67T^{2} \)
71 \( 1 + 4.38T + 71T^{2} \)
73 \( 1 + 3.89iT - 73T^{2} \)
79 \( 1 - 8.94iT - 79T^{2} \)
83 \( 1 + 13.1iT - 83T^{2} \)
89 \( 1 + 7.51iT - 89T^{2} \)
97 \( 1 + 18.3T + 97T^{2} \)
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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

−9.506386998814407897339893574094, −9.023856755300595765172712121967, −8.415210397719980522725307196198, −7.57416069254016773573458598350, −7.26841103241458742525260945386, −5.84721541292742935314337607843, −4.72608882971104081112405049383, −3.27878280808338789334147103393, −1.96349350724624955622377134348, −1.17247316749384915376134328676, 0.75784907521044015751849332471, 2.04536808778532028782682573190, 3.10074430685865390310036552227, 4.17638333459188630223427221282, 6.04438210105856793668759216859, 6.91392701477540048783441887894, 7.49453890824005415536423929216, 8.241855610580235426814725986823, 8.732912503684278357648989243521, 9.646856872124017862312444883815

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