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Properties

Label 2-1183-13.12-c1-0-42
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  + 1.81i·2-s − 3.10·3-s − 1.28·4-s − 2.81i·5-s − 5.62i·6-s − i·7-s + 1.28i·8-s + 6.62·9-s + 5.10·10-s + 3.10i·11-s + 3.99·12-s + 1.81·14-s + 8.72i·15-s − 4.91·16-s + 0.524·17-s + 12.0i·18-s + ⋯
L(s)  = 1  + 1.28i·2-s − 1.79·3-s − 0.644·4-s − 1.25i·5-s − 2.29i·6-s − 0.377i·7-s + 0.455i·8-s + 2.20·9-s + 1.61·10-s + 0.935i·11-s + 1.15·12-s + 0.484·14-s + 2.25i·15-s − 1.22·16-s + 0.127·17-s + 2.83i·18-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

\(L(1)\) \(\approx\) \(0.3869067374\)
\(L(\frac12)\) \(\approx\) \(0.3869067374\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.81iT - 2T^{2} \)
3 \( 1 + 3.10T + 3T^{2} \)
5 \( 1 + 2.81iT - 5T^{2} \)
11 \( 1 - 3.10iT - 11T^{2} \)
17 \( 1 - 0.524T + 17T^{2} \)
19 \( 1 + 0.813iT - 19T^{2} \)
23 \( 1 + 7.33T + 23T^{2} \)
29 \( 1 - 8.28T + 29T^{2} \)
31 \( 1 + 1.39iT - 31T^{2} \)
37 \( 1 + 6.15iT - 37T^{2} \)
41 \( 1 - 4.20iT - 41T^{2} \)
43 \( 1 + 6.75T + 43T^{2} \)
47 \( 1 + 5.97iT - 47T^{2} \)
53 \( 1 + 2.49T + 53T^{2} \)
59 \( 1 + 4.47iT - 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 10.0iT - 67T^{2} \)
71 \( 1 - 8.72iT - 71T^{2} \)
73 \( 1 + 2.34iT - 73T^{2} \)
79 \( 1 + 13.5T + 79T^{2} \)
83 \( 1 + 16.4iT - 83T^{2} \)
89 \( 1 + 10.6iT - 89T^{2} \)
97 \( 1 - 1.18iT - 97T^{2} \)
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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

−9.706206763570249181937989237867, −8.568978821176591726459614112767, −7.73935971739263477136741828596, −6.93498936184320894009913924832, −6.23912587894301366214357941256, −5.50809280604860920434196333834, −4.76189370129714084408958254134, −4.34016419538230622065122289447, −1.68406730606267219456071807991, −0.23376255480515898519890842593, 1.23147116243437430192303688059, 2.59969892386471425594724065270, 3.58693840095587296207611391122, 4.63112652596619947596383855799, 5.81194575700257590348402081380, 6.39640125927720264796381543172, 7.04158028114937041589435592693, 8.333502336478287549475981881424, 9.800507972890337218129799855784, 10.23434295627926077322781690443

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