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Properties

Label 2-1183-13.12-c1-0-58
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  + 2.23i·2-s + 3.02·3-s − 3.00·4-s − 3.28i·5-s + 6.76i·6-s − i·7-s − 2.24i·8-s + 6.15·9-s + 7.35·10-s − 3.69i·11-s − 9.09·12-s + 2.23·14-s − 9.94i·15-s − 0.979·16-s − 0.705·17-s + 13.7i·18-s + ⋯
L(s)  = 1  + 1.58i·2-s + 1.74·3-s − 1.50·4-s − 1.46i·5-s + 2.76i·6-s − 0.377i·7-s − 0.795i·8-s + 2.05·9-s + 2.32·10-s − 1.11i·11-s − 2.62·12-s + 0.597·14-s − 2.56i·15-s − 0.244·16-s − 0.171·17-s + 3.24i·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

\(L(1)\) \(\approx\) \(2.845481337\)
\(L(\frac12)\) \(\approx\) \(2.845481337\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.23iT - 2T^{2} \)
3 \( 1 - 3.02T + 3T^{2} \)
5 \( 1 + 3.28iT - 5T^{2} \)
11 \( 1 + 3.69iT - 11T^{2} \)
17 \( 1 + 0.705T + 17T^{2} \)
19 \( 1 - 0.911iT - 19T^{2} \)
23 \( 1 - 3.01T + 23T^{2} \)
29 \( 1 - 6.55T + 29T^{2} \)
31 \( 1 + 9.55iT - 31T^{2} \)
37 \( 1 - 7.26iT - 37T^{2} \)
41 \( 1 - 0.884iT - 41T^{2} \)
43 \( 1 + 0.536T + 43T^{2} \)
47 \( 1 - 11.4iT - 47T^{2} \)
53 \( 1 - 3.77T + 53T^{2} \)
59 \( 1 - 7.40iT - 59T^{2} \)
61 \( 1 + 1.81T + 61T^{2} \)
67 \( 1 - 6.41iT - 67T^{2} \)
71 \( 1 - 10.7iT - 71T^{2} \)
73 \( 1 + 9.72iT - 73T^{2} \)
79 \( 1 + 7.00T + 79T^{2} \)
83 \( 1 + 2.31iT - 83T^{2} \)
89 \( 1 - 2.23iT - 89T^{2} \)
97 \( 1 - 8.79iT - 97T^{2} \)
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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

−9.157353260017350243351530632856, −8.856400858639772732332904845092, −8.076441430933230114091324858293, −7.83636366570707973298063288067, −6.71771580463878108245959931559, −5.73731685828329362258458959656, −4.67538518302721896878816590602, −4.06490884690095363207252464684, −2.76888718405148706893869822393, −1.10346524570976729764368635440, 1.76757765642008830840857130037, 2.49117054071159887645442235439, 3.08496371595955341133042762178, 3.79790878516588089829031982981, 4.82936811631106616381703027435, 6.79278840408855132307386679544, 7.20752778293685053879950018907, 8.398444637179412861750406249358, 9.084946227615374499122279406954, 9.839177802334475379798093688432

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