Properties

Label 2-1183-13.12-c1-0-12
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 + 4.41i·5-s − 2.00i·6-s i·7-s + 2.82i·8-s − 0.999·9-s − 6.24·10-s + 4.24i·11-s + 1.41·14-s − 6.24i·15-s − 4.00·16-s + 1.41·17-s − 1.41i·18-s + 1.24i·19-s + ⋯
L(s)  = 1  + 0.999i·2-s − 0.816·3-s + 1.97i·5-s − 0.816i·6-s − 0.377i·7-s + 0.999i·8-s − 0.333·9-s − 1.97·10-s + 1.27i·11-s + 0.377·14-s − 1.61i·15-s − 1.00·16-s + 0.342·17-s − 0.333i·18-s + 0.285i·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

\(L(1)\) \(\approx\) \(1.002161129\)
\(L(\frac12)\) \(\approx\) \(1.002161129\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.41iT - 5T^{2} \)
11 \( 1 - 4.24iT - 11T^{2} \)
17 \( 1 - 1.41T + 17T^{2} \)
19 \( 1 - 1.24iT - 19T^{2} \)
23 \( 1 - 0.171T + 23T^{2} \)
29 \( 1 - 5.82T + 29T^{2} \)
31 \( 1 + 5.24iT - 31T^{2} \)
37 \( 1 - 6.24iT - 37T^{2} \)
41 \( 1 - 3.17iT - 41T^{2} \)
43 \( 1 - 5T + 43T^{2} \)
47 \( 1 + 4.41iT - 47T^{2} \)
53 \( 1 + 5.82T + 53T^{2} \)
59 \( 1 + 11.6iT - 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 - 2.48iT - 67T^{2} \)
71 \( 1 - 1.07iT - 71T^{2} \)
73 \( 1 - 0.757iT - 73T^{2} \)
79 \( 1 + 1.48T + 79T^{2} \)
83 \( 1 - 4.75iT - 83T^{2} \)
89 \( 1 + 4.41iT - 89T^{2} \)
97 \( 1 + 13.7iT - 97T^{2} \)
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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

−10.30388306205652173772524308249, −9.791497240452510818623335349653, −8.209891791011899508975360580086, −7.50963557485982596358093243374, −6.80593200785392358966371295315, −6.39768233294012560897383534203, −5.61711604944234924116434855354, −4.53016237893200899953251155818, −3.14887016743453032469639274128, −2.18203868392107879746775730904, 0.50298107073388121815470640618, 1.29139106747678635455856508547, 2.74061153298549914710427263892, 3.93484271803200795199379939235, 4.96868573668115840494075677370, 5.64853956109799753822711512629, 6.38162476015915415516721169592, 7.83418859692572087571015860634, 8.797977363947658766423627176839, 9.112362311205722909600983456365

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