Properties

Label 2-1183-1.1-c1-0-43
Degree $2$
Conductor $1183$
Sign $1$
Analytic cond. $9.44630$
Root an. cond. $3.07348$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.149·2-s + 2.76·3-s − 1.97·4-s + 4.13·5-s − 0.413·6-s + 7-s + 0.595·8-s + 4.61·9-s − 0.618·10-s − 2.55·11-s − 5.45·12-s − 0.149·14-s + 11.4·15-s + 3.86·16-s + 1.50·17-s − 0.691·18-s − 5.93·19-s − 8.17·20-s + 2.76·21-s + 0.382·22-s + 6.55·23-s + 1.64·24-s + 12.0·25-s + 4.46·27-s − 1.97·28-s + 0.283·29-s − 1.70·30-s + ⋯
L(s)  = 1  − 0.105·2-s + 1.59·3-s − 0.988·4-s + 1.84·5-s − 0.168·6-s + 0.377·7-s + 0.210·8-s + 1.53·9-s − 0.195·10-s − 0.769·11-s − 1.57·12-s − 0.0399·14-s + 2.94·15-s + 0.966·16-s + 0.364·17-s − 0.162·18-s − 1.36·19-s − 1.82·20-s + 0.602·21-s + 0.0814·22-s + 1.36·23-s + 0.335·24-s + 2.41·25-s + 0.860·27-s − 0.373·28-s + 0.0527·29-s − 0.311·30-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(9.44630\)
Root analytic conductor: \(3.07348\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.008960736\)
\(L(\frac12)\) \(\approx\) \(3.008960736\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 - T \)
13 \( 1 \)
good2 \( 1 + 0.149T + 2T^{2} \)
3 \( 1 - 2.76T + 3T^{2} \)
5 \( 1 - 4.13T + 5T^{2} \)
11 \( 1 + 2.55T + 11T^{2} \)
17 \( 1 - 1.50T + 17T^{2} \)
19 \( 1 + 5.93T + 19T^{2} \)
23 \( 1 - 6.55T + 23T^{2} \)
29 \( 1 - 0.283T + 29T^{2} \)
31 \( 1 + 1.95T + 31T^{2} \)
37 \( 1 - 5.66T + 37T^{2} \)
41 \( 1 + 6.70T + 41T^{2} \)
43 \( 1 + 8.14T + 43T^{2} \)
47 \( 1 + 3.94T + 47T^{2} \)
53 \( 1 + 1.08T + 53T^{2} \)
59 \( 1 + 3.71T + 59T^{2} \)
61 \( 1 - 1.93T + 61T^{2} \)
67 \( 1 + 3.38T + 67T^{2} \)
71 \( 1 - 5.36T + 71T^{2} \)
73 \( 1 - 2.62T + 73T^{2} \)
79 \( 1 - 7.89T + 79T^{2} \)
83 \( 1 + 10.5T + 83T^{2} \)
89 \( 1 + 6.64T + 89T^{2} \)
97 \( 1 - 0.504T + 97T^{2} \)
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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

−9.722352293408528770416247115427, −8.800223968277557370400423614857, −8.561041893781564363550256719873, −7.58971199213378729477828910720, −6.44725768772857573217026520487, −5.33130155510531160222778668624, −4.65908207788150355938612073010, −3.33468359756014898526817254146, −2.41941898987002077684020922232, −1.50312160662330544905113176583, 1.50312160662330544905113176583, 2.41941898987002077684020922232, 3.33468359756014898526817254146, 4.65908207788150355938612073010, 5.33130155510531160222778668624, 6.44725768772857573217026520487, 7.58971199213378729477828910720, 8.561041893781564363550256719873, 8.800223968277557370400423614857, 9.722352293408528770416247115427

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