↑

Properties

Label 2-1183-13.12-c1-0-69
Degree $2$
Conductor $1183$
Sign $0.722 - 0.691i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.07i·2-s − 3.01·3-s − 2.29·4-s − 2.69i·5-s + 6.24i·6-s − i·7-s + 0.602i·8-s + 6.09·9-s − 5.57·10-s − 1.66i·11-s + 6.90·12-s − 2.07·14-s + 8.12i·15-s − 3.33·16-s − 6.90·17-s − 12.6i·18-s + ⋯
L(s)  = 1  − 1.46i·2-s − 1.74·3-s − 1.14·4-s − 1.20i·5-s + 2.55i·6-s − 0.377i·7-s + 0.212i·8-s + 2.03·9-s − 1.76·10-s − 0.500i·11-s + 1.99·12-s − 0.553·14-s + 2.09i·15-s − 0.833·16-s − 1.67·17-s − 2.97i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 - 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.722 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.722 - 0.691i$
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: \((2,\ 1183,\ (\ :1/2),\ 0.722 - 0.691i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3997204587\)
\(L(\frac12)\) \(\approx\) \(0.3997204587\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.07iT - 2T^{2} \)
3 \( 1 + 3.01T + 3T^{2} \)
5 \( 1 + 2.69iT - 5T^{2} \)
11 \( 1 + 1.66iT - 11T^{2} \)
17 \( 1 + 6.90T + 17T^{2} \)
19 \( 1 + 7.92iT - 19T^{2} \)
23 \( 1 + 1.95T + 23T^{2} \)
29 \( 1 + 2.71T + 29T^{2} \)
31 \( 1 + 1.76iT - 31T^{2} \)
37 \( 1 - 4.20iT - 37T^{2} \)
41 \( 1 + 7.08iT - 41T^{2} \)
43 \( 1 - 3.56T + 43T^{2} \)
47 \( 1 - 1.53iT - 47T^{2} \)
53 \( 1 - 12.7T + 53T^{2} \)
59 \( 1 + 4.52iT - 59T^{2} \)
61 \( 1 - 2.89T + 61T^{2} \)
67 \( 1 + 1.27iT - 67T^{2} \)
71 \( 1 + 2.93iT - 71T^{2} \)
73 \( 1 - 14.6iT - 73T^{2} \)
79 \( 1 - 2.91T + 79T^{2} \)
83 \( 1 + 2.33iT - 83T^{2} \)
89 \( 1 + 7.64iT - 89T^{2} \)
97 \( 1 - 12.9iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.259539867115564726243505865364, −8.721151808957957692319255073212, −7.15183435521139099776297239292, −6.40600914742749253900177150434, −5.26492465572969062418549500523, −4.62377390032039501563622820914, −3.98184224637067603633561516019, −2.26822323827358011443944482382, −0.957269610399379947491762410554, −0.27457620266479108521417433051, 2.11568856900857435721134990217, 4.01379039827526128380476760622, 4.91903547833709846972045738324, 5.90241361552748089578865169011, 6.19627956966459753559262381173, 6.99063787950632844011717328173, 7.48364602015936470580972412055, 8.618472033903888609025048455307, 9.779782929295500500007233856317, 10.55832425778154558388253398492

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