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Properties

Label 2-1183-13.12-c1-0-52
Degree $2$
Conductor $1183$
Sign $0.960 + 0.277i$
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  − 0.823i·2-s + 2.66·3-s + 1.32·4-s + 3.16i·5-s − 2.19i·6-s + i·7-s − 2.73i·8-s + 4.07·9-s + 2.60·10-s − 5.94i·11-s + 3.51·12-s + 0.823·14-s + 8.41i·15-s + 0.390·16-s + 2.69·17-s − 3.35i·18-s + ⋯
L(s)  = 1  − 0.582i·2-s + 1.53·3-s + 0.660·4-s + 1.41i·5-s − 0.894i·6-s + 0.377i·7-s − 0.967i·8-s + 1.35·9-s + 0.823·10-s − 1.79i·11-s + 1.01·12-s + 0.220·14-s + 2.17i·15-s + 0.0976·16-s + 0.654·17-s − 0.791i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.960 + 0.277i$
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: \((2,\ 1183,\ (\ :1/2),\ 0.960 + 0.277i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.384573689\)
\(L(\frac12)\) \(\approx\) \(3.384573689\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.823iT - 2T^{2} \)
3 \( 1 - 2.66T + 3T^{2} \)
5 \( 1 - 3.16iT - 5T^{2} \)
11 \( 1 + 5.94iT - 11T^{2} \)
17 \( 1 - 2.69T + 17T^{2} \)
19 \( 1 - 1.95iT - 19T^{2} \)
23 \( 1 - 2.72T + 23T^{2} \)
29 \( 1 + 5.99T + 29T^{2} \)
31 \( 1 - 1.15iT - 31T^{2} \)
37 \( 1 - 6.50iT - 37T^{2} \)
41 \( 1 - 3.73iT - 41T^{2} \)
43 \( 1 + 6.99T + 43T^{2} \)
47 \( 1 + 0.456iT - 47T^{2} \)
53 \( 1 - 0.399T + 53T^{2} \)
59 \( 1 - 4.80iT - 59T^{2} \)
61 \( 1 + 1.15T + 61T^{2} \)
67 \( 1 - 6.27iT - 67T^{2} \)
71 \( 1 - 4.50iT - 71T^{2} \)
73 \( 1 + 8.30iT - 73T^{2} \)
79 \( 1 + 7.91T + 79T^{2} \)
83 \( 1 + 6.19iT - 83T^{2} \)
89 \( 1 + 3.56iT - 89T^{2} \)
97 \( 1 + 3.42iT - 97T^{2} \)
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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

−9.866387944460672262023297333598, −8.909621459469738604146201598459, −8.117557845716181289150931317769, −7.39898946277058318305460980289, −6.54550846557838991521615861845, −5.72697401561464941937952252077, −3.75930021145887242785015074534, −3.08913387249121934505283198397, −2.84162298418150619930541691132, −1.60563026683544439854038155034, 1.55553770622918272157799088115, 2.33701501286827731354639171623, 3.65664772775856757951121507682, 4.65738640409897968467332088068, 5.43098591373895047085905248527, 6.87380228017276060381298518331, 7.52273783988997959432055512408, 8.026213783586055278153903887324, 8.939896328914450230497059128129, 9.449638245777698463510540750906

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