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Properties

Label 2-1183-13.12-c1-0-16
Degree $2$
Conductor $1183$
Sign $-0.0304 - 0.999i$
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  + 0.961i·2-s − 1.98·3-s + 1.07·4-s − 3.39i·5-s − 1.91i·6-s + i·7-s + 2.95i·8-s + 0.949·9-s + 3.26·10-s + 4.59i·11-s − 2.13·12-s − 0.961·14-s + 6.74i·15-s − 0.692·16-s − 2.44·17-s + 0.913i·18-s + ⋯
L(s)  = 1  + 0.679i·2-s − 1.14·3-s + 0.537·4-s − 1.51i·5-s − 0.780i·6-s + 0.377i·7-s + 1.04i·8-s + 0.316·9-s + 1.03·10-s + 1.38i·11-s − 0.616·12-s − 0.256·14-s + 1.74i·15-s − 0.173·16-s − 0.593·17-s + 0.215i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0304 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0304 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.0304 - 0.999i$
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: \((2,\ 1183,\ (\ :1/2),\ -0.0304 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.072593436\)
\(L(\frac12)\) \(\approx\) \(1.072593436\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.961iT - 2T^{2} \)
3 \( 1 + 1.98T + 3T^{2} \)
5 \( 1 + 3.39iT - 5T^{2} \)
11 \( 1 - 4.59iT - 11T^{2} \)
17 \( 1 + 2.44T + 17T^{2} \)
19 \( 1 + 4.77iT - 19T^{2} \)
23 \( 1 - 4.04T + 23T^{2} \)
29 \( 1 + 3.20T + 29T^{2} \)
31 \( 1 - 4.83iT - 31T^{2} \)
37 \( 1 - 9.61iT - 37T^{2} \)
41 \( 1 - 8.99iT - 41T^{2} \)
43 \( 1 - 8.90T + 43T^{2} \)
47 \( 1 + 5.37iT - 47T^{2} \)
53 \( 1 - 9.97T + 53T^{2} \)
59 \( 1 - 10.5iT - 59T^{2} \)
61 \( 1 - 5.29T + 61T^{2} \)
67 \( 1 + 14.0iT - 67T^{2} \)
71 \( 1 - 8.52iT - 71T^{2} \)
73 \( 1 - 6.62iT - 73T^{2} \)
79 \( 1 + 7.98T + 79T^{2} \)
83 \( 1 - 7.30iT - 83T^{2} \)
89 \( 1 - 18.1iT - 89T^{2} \)
97 \( 1 + 6.90iT - 97T^{2} \)
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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

−9.914714035471639806621611906229, −8.962931409625899074846843807656, −8.382693917469709170704507483196, −7.21917440427717039731823481287, −6.65820734106176009700594966795, −5.66196151621352427064472024008, −5.00326976528227675083872650143, −4.55996532747053056625287001023, −2.51490841102168502936499160242, −1.20242887293565371671717829333, 0.58347428972941575067386389089, 2.23629406851997470284225477991, 3.24024561951210353757347147736, 3.99536793371847114224158460429, 5.75760401548374833502190409492, 6.04894002827192448178363792813, 6.98122794414029939417205055537, 7.54414161128338471701889308024, 8.923772114273669570888201714213, 10.13325051491740146487140849946

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