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Properties

Label 2-366-61.47-c1-0-0
Degree $2$
Conductor $366$
Sign $-0.614 - 0.789i$
Analytic cond. $2.92252$
Root an. cond. $1.70953$
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  + (0.5 + 0.866i)2-s − 3-s + (−0.499 + 0.866i)4-s + (2.19 + 3.79i)5-s + (−0.5 − 0.866i)6-s + (−0.135 − 0.234i)7-s − 0.999·8-s + 9-s + (−2.19 + 3.79i)10-s − 0.270·11-s + (0.499 − 0.866i)12-s + (−1.27 − 2.20i)13-s + (0.135 − 0.234i)14-s + (−2.19 − 3.79i)15-s + (−0.5 − 0.866i)16-s + (−3.42 + 5.92i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s − 0.577·3-s + (−0.249 + 0.433i)4-s + (0.980 + 1.69i)5-s + (−0.204 − 0.353i)6-s + (−0.0511 − 0.0886i)7-s − 0.353·8-s + 0.333·9-s + (−0.693 + 1.20i)10-s − 0.0816·11-s + (0.144 − 0.249i)12-s + (−0.352 − 0.610i)13-s + (0.0361 − 0.0626i)14-s + (−0.566 − 0.980i)15-s + (−0.125 − 0.216i)16-s + (−0.829 + 1.43i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.614 - 0.789i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.614 - 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(366\)    =    \(2 \cdot 3 \cdot 61\)
Sign: $-0.614 - 0.789i$
Analytic conductor: \(2.92252\)
Root analytic conductor: \(1.70953\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{366} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 366,\ (\ :1/2),\ -0.614 - 0.789i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.616926 + 1.26171i\)
\(L(\frac12)\) \(\approx\) \(0.616926 + 1.26171i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + T \)
61 \( 1 + (-7.17 - 3.09i)T \)
good5 \( 1 + (-2.19 - 3.79i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.135 + 0.234i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 0.270T + 11T^{2} \)
13 \( 1 + (1.27 + 2.20i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.42 - 5.92i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 5.11T + 23T^{2} \)
29 \( 1 + (-0.229 + 0.396i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.11T + 37T^{2} \)
41 \( 1 - 2.72T + 41T^{2} \)
43 \( 1 + (-4.82 - 8.36i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.94 + 8.56i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 8.61T + 53T^{2} \)
59 \( 1 + (5.38 + 9.32i)T + (-29.5 + 51.0i)T^{2} \)
67 \( 1 + (-1.55 - 2.69i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.729 + 1.26i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.5 + 4.33i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.84 + 6.65i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.86 + 3.22i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 13.7T + 89T^{2} \)
97 \( 1 + (-3.07 + 5.33i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.59711143097059488862479245753, −10.67656312024268872347107309141, −10.21500934750857306971733249691, −9.014858861002340517890092113886, −7.56783690411986937432772721642, −6.75976730989470201309205422865, −6.12209682195960143587886093672, −5.20062893242564309661969009716, −3.63700910620987928193226476679, −2.37000302787835444948396067627, 0.959667105738758616862177625672, 2.33604205950120538196913605650, 4.36973121353154548142364689998, 5.07283175500246624142591699511, 5.81810472280475581351845273222, 7.12065351943739252519272864986, 8.825684818222730411797383662559, 9.254709291459052360748344957121, 10.14872279371118090352264822472, 11.25645018488591354390139217763

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