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Properties

Label 2-366-61.13-c1-0-5
Degree $2$
Conductor $366$
Sign $0.255 + 0.966i$
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 + (−1.74 + 3.02i)5-s + (−0.5 + 0.866i)6-s + (2.04 − 3.53i)7-s − 0.999·8-s + 9-s + (1.74 + 3.02i)10-s + 4.08·11-s + (0.499 + 0.866i)12-s + (3.08 − 5.33i)13-s + (−2.04 − 3.53i)14-s + (1.74 − 3.02i)15-s + (−0.5 + 0.866i)16-s + (−3.83 − 6.64i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s − 0.577·3-s + (−0.249 − 0.433i)4-s + (−0.780 + 1.35i)5-s + (−0.204 + 0.353i)6-s + (0.771 − 1.33i)7-s − 0.353·8-s + 0.333·9-s + (0.551 + 0.955i)10-s + 1.23·11-s + (0.144 + 0.249i)12-s + (0.854 − 1.47i)13-s + (−0.545 − 0.944i)14-s + (0.450 − 0.780i)15-s + (−0.125 + 0.216i)16-s + (−0.930 − 1.61i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.255 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.255 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(366\)    =    \(2 \cdot 3 \cdot 61\)
Sign: $0.255 + 0.966i$
Analytic conductor: \(2.92252\)
Root analytic conductor: \(1.70953\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{366} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 366,\ (\ :1/2),\ 0.255 + 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01394 - 0.780773i\)
\(L(\frac12)\) \(\approx\) \(1.01394 - 0.780773i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.88 + 7.57i)T \)
good5 \( 1 + (1.74 - 3.02i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2.04 + 3.53i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 4.08T + 11T^{2} \)
13 \( 1 + (-3.08 + 5.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.83 + 6.64i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 1.59T + 23T^{2} \)
29 \( 1 + (-4.58 - 7.93i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 2.59T + 37T^{2} \)
41 \( 1 - 7.08T + 41T^{2} \)
43 \( 1 + (1.28 - 2.22i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.69 + 8.13i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 6.30T + 53T^{2} \)
59 \( 1 + (-2.49 + 4.31i)T + (-29.5 - 51.0i)T^{2} \)
67 \( 1 + (0.204 - 0.354i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.08 - 8.79i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.5 - 4.33i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.67 - 8.09i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.04 - 6.99i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 15.8T + 89T^{2} \)
97 \( 1 + (-2.66 - 4.61i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.06226236907451347449265835146, −10.83440129384430569778741599980, −9.848129090766527381068675983526, −8.315108618785523235353235887839, −7.15110907813220437291870109792, −6.64151957066115084756796554573, −5.07942057319333933364028445653, −3.97692576167411312892350600542, −3.14272014261601278187200170090, −0.985782425999355946798627676445, 1.61121903139439516655386133868, 4.19936781175274643038305336093, 4.52444475357406099646555038591, 5.85041306155827942672045660020, 6.53606426263852018214995564663, 8.065616019997167754424440484929, 8.776017942831535473935181055689, 9.213375654867571259403888807819, 11.25802714273247170837101090097, 11.72056815726889972161083978427

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