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Properties

Label 2-1183-13.12-c1-0-62
Degree $2$
Conductor $1183$
Sign $-0.554 + 0.832i$
Analytic cond. $9.44630$
Root an. cond. $3.07348$
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  − 1.41i·2-s + 1.41·3-s + 1.58i·5-s − 2.00i·6-s − i·7-s − 2.82i·8-s − 0.999·9-s + 2.24·10-s − 4.24i·11-s − 1.41·14-s + 2.24i·15-s − 4.00·16-s − 1.41·17-s + 1.41i·18-s − 7.24i·19-s + ⋯
L(s)  = 1  − 0.999i·2-s + 0.816·3-s + 0.709i·5-s − 0.816i·6-s − 0.377i·7-s − 0.999i·8-s − 0.333·9-s + 0.709·10-s − 1.27i·11-s − 0.377·14-s + 0.579i·15-s − 1.00·16-s − 0.342·17-s + 0.333i·18-s − 1.66i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.554 + 0.832i$
Analytic conductor: \(9.44630\)
Root analytic conductor: \(3.07348\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :1/2),\ -0.554 + 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.180482680\)
\(L(\frac12)\) \(\approx\) \(2.180482680\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + iT \)
13 \( 1 \)
good2 \( 1 + 1.41iT - 2T^{2} \)
3 \( 1 - 1.41T + 3T^{2} \)
5 \( 1 - 1.58iT - 5T^{2} \)
11 \( 1 + 4.24iT - 11T^{2} \)
17 \( 1 + 1.41T + 17T^{2} \)
19 \( 1 + 7.24iT - 19T^{2} \)
23 \( 1 - 5.82T + 23T^{2} \)
29 \( 1 - 0.171T + 29T^{2} \)
31 \( 1 - 3.24iT - 31T^{2} \)
37 \( 1 + 2.24iT - 37T^{2} \)
41 \( 1 - 8.82iT - 41T^{2} \)
43 \( 1 - 5T + 43T^{2} \)
47 \( 1 + 1.58iT - 47T^{2} \)
53 \( 1 + 0.171T + 53T^{2} \)
59 \( 1 + 0.343iT - 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 + 14.4iT - 67T^{2} \)
71 \( 1 + 13.0iT - 71T^{2} \)
73 \( 1 - 9.24iT - 73T^{2} \)
79 \( 1 - 15.4T + 79T^{2} \)
83 \( 1 - 13.2iT - 83T^{2} \)
89 \( 1 + 1.58iT - 89T^{2} \)
97 \( 1 - 11.7iT - 97T^{2} \)
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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

−9.401723757013046224399742046152, −8.960923925744453755647787360595, −7.968203209682079125586698920951, −6.95604667606353806088040006308, −6.38809563866148815457226451948, −4.98004940742557123707613387922, −3.65361703743819101139162561290, −3.00611093591508705315434933700, −2.45600496581358883516590133258, −0.837244320492545906829699994381, 1.79555544560301329292161094314, 2.73768468888643032198507652173, 4.11483846106220186376339103959, 5.15702037904499284917634074172, 5.82586967476106075460732064030, 6.90849367139665123020828639816, 7.61991663654336066393444420439, 8.381403130520150635558469010734, 8.911118635912522978912501687865, 9.682213235060810957149366448704

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