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Properties

Label 2-1183-13.12-c1-0-1
Degree $2$
Conductor $1183$
Sign $-0.969 + 0.246i$
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  + 2.62i·2-s − 1.76·3-s − 4.90·4-s − 2.94i·5-s − 4.64i·6-s − i·7-s − 7.63i·8-s + 0.124·9-s + 7.73·10-s − 4.28i·11-s + 8.67·12-s + 2.62·14-s + 5.20i·15-s + 10.2·16-s − 2.94·17-s + 0.327i·18-s + ⋯
L(s)  = 1  + 1.85i·2-s − 1.02·3-s − 2.45·4-s − 1.31i·5-s − 1.89i·6-s − 0.377i·7-s − 2.70i·8-s + 0.0414·9-s + 2.44·10-s − 1.29i·11-s + 2.50·12-s + 0.702·14-s + 1.34i·15-s + 2.56·16-s − 0.713·17-s + 0.0770i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.246i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.969 + 0.246i$
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: \((2,\ 1183,\ (\ :1/2),\ -0.969 + 0.246i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2907676868\)
\(L(\frac12)\) \(\approx\) \(0.2907676868\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.62iT - 2T^{2} \)
3 \( 1 + 1.76T + 3T^{2} \)
5 \( 1 + 2.94iT - 5T^{2} \)
11 \( 1 + 4.28iT - 11T^{2} \)
17 \( 1 + 2.94T + 17T^{2} \)
19 \( 1 - 4.98iT - 19T^{2} \)
23 \( 1 + 7.58T + 23T^{2} \)
29 \( 1 + 1.82T + 29T^{2} \)
31 \( 1 - 6.57iT - 31T^{2} \)
37 \( 1 - 8.31iT - 37T^{2} \)
41 \( 1 + 4.15iT - 41T^{2} \)
43 \( 1 - 8.48T + 43T^{2} \)
47 \( 1 + 0.910iT - 47T^{2} \)
53 \( 1 - 4.47T + 53T^{2} \)
59 \( 1 - 9.42iT - 59T^{2} \)
61 \( 1 + 1.75T + 61T^{2} \)
67 \( 1 - 0.413iT - 67T^{2} \)
71 \( 1 + 2.95iT - 71T^{2} \)
73 \( 1 - 0.885iT - 73T^{2} \)
79 \( 1 - 10.0T + 79T^{2} \)
83 \( 1 - 6.04iT - 83T^{2} \)
89 \( 1 - 3.82iT - 89T^{2} \)
97 \( 1 + 2.37iT - 97T^{2} \)
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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

−9.964740609194918955948328321031, −8.881711270883846090218815108866, −8.471959946112952414604085082588, −7.77209521278717532818945944167, −6.63345823586170867827615532882, −5.92136204429335282213704643679, −5.48372464153074442024655095618, −4.65160845178766350540822303441, −3.82400470277787433807938916775, −0.955899320300749518619029290087, 0.18669103132164260941522911779, 2.10819838516439670601962225067, 2.62603071920253255836122016316, 3.92027999206024391970483275407, 4.70018930346191770813733836983, 5.75209938979443216162411885016, 6.68131098854234505076373476467, 7.73710782596747325868742273784, 9.032562442232638677980096411512, 9.725416195135094256227347267168

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