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Properties

Label 2-1183-13.12-c1-0-49
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.149i·2-s + 2.76·3-s + 1.97·4-s + 4.13i·5-s − 0.413i·6-s − i·7-s − 0.595i·8-s + 4.61·9-s + 0.618·10-s + 2.55i·11-s + 5.45·12-s − 0.149·14-s + 11.4i·15-s + 3.86·16-s − 1.50·17-s − 0.691i·18-s + ⋯
L(s)  = 1  − 0.105i·2-s + 1.59·3-s + 0.988·4-s + 1.84i·5-s − 0.168i·6-s − 0.377i·7-s − 0.210i·8-s + 1.53·9-s + 0.195·10-s + 0.769i·11-s + 1.57·12-s − 0.0399·14-s + 2.94i·15-s + 0.966·16-s − 0.364·17-s − 0.162i·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

\(L(1)\) \(\approx\) \(3.479606615\)
\(L(\frac12)\) \(\approx\) \(3.479606615\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.149iT - 2T^{2} \)
3 \( 1 - 2.76T + 3T^{2} \)
5 \( 1 - 4.13iT - 5T^{2} \)
11 \( 1 - 2.55iT - 11T^{2} \)
17 \( 1 + 1.50T + 17T^{2} \)
19 \( 1 + 5.93iT - 19T^{2} \)
23 \( 1 + 6.55T + 23T^{2} \)
29 \( 1 - 0.283T + 29T^{2} \)
31 \( 1 + 1.95iT - 31T^{2} \)
37 \( 1 + 5.66iT - 37T^{2} \)
41 \( 1 + 6.70iT - 41T^{2} \)
43 \( 1 - 8.14T + 43T^{2} \)
47 \( 1 - 3.94iT - 47T^{2} \)
53 \( 1 + 1.08T + 53T^{2} \)
59 \( 1 - 3.71iT - 59T^{2} \)
61 \( 1 - 1.93T + 61T^{2} \)
67 \( 1 + 3.38iT - 67T^{2} \)
71 \( 1 - 5.36iT - 71T^{2} \)
73 \( 1 + 2.62iT - 73T^{2} \)
79 \( 1 - 7.89T + 79T^{2} \)
83 \( 1 + 10.5iT - 83T^{2} \)
89 \( 1 - 6.64iT - 89T^{2} \)
97 \( 1 - 0.504iT - 97T^{2} \)
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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

−9.926712548473340648731981782464, −9.177223084706835188647483102393, −7.912147658294991219985834108674, −7.36666081774686506000211748919, −6.92727048988158527465841550300, −6.03910535040330605852399947731, −4.15504092407431979811326667598, −3.39945439270224790847016242865, −2.44500561719935921308275660884, −2.15841324931164942804046852482, 1.38802875215899210107331621423, 2.20147294258284861136431629560, 3.38198863691151545591524597367, 4.25930735406136291027524222629, 5.48321798323146671787709539949, 6.25887107245756455092538778699, 7.65948581388541983394153398303, 8.249956034569618938604169514227, 8.544124550528839889558587744585, 9.486734374020988338081569451638

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