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Properties

Label 2-366-183.104-c1-0-6
Degree $2$
Conductor $366$
Sign $0.977 - 0.210i$
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  + (−0.777 − 0.629i)2-s + (1.66 + 0.480i)3-s + (0.207 + 0.978i)4-s + (0.665 + 0.739i)5-s + (−0.990 − 1.42i)6-s + (2.47 + 0.949i)7-s + (0.453 − 0.891i)8-s + (2.53 + 1.60i)9-s + (−0.0520 − 0.993i)10-s + (−2.73 − 2.73i)11-s + (−0.124 + 1.72i)12-s + (−1.80 + 3.13i)13-s + (−1.32 − 2.29i)14-s + (0.752 + 1.55i)15-s + (−0.913 + 0.406i)16-s + (2.77 + 1.80i)17-s + ⋯
L(s)  = 1  + (−0.549 − 0.444i)2-s + (0.960 + 0.277i)3-s + (0.103 + 0.489i)4-s + (0.297 + 0.330i)5-s + (−0.404 − 0.580i)6-s + (0.934 + 0.358i)7-s + (0.160 − 0.315i)8-s + (0.845 + 0.533i)9-s + (−0.0164 − 0.314i)10-s + (−0.823 − 0.823i)11-s + (−0.0359 + 0.498i)12-s + (−0.501 + 0.869i)13-s + (−0.354 − 0.613i)14-s + (0.194 + 0.400i)15-s + (−0.228 + 0.101i)16-s + (0.673 + 0.437i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.210i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 - 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(366\)    =    \(2 \cdot 3 \cdot 61\)
Sign: $0.977 - 0.210i$
Analytic conductor: \(2.92252\)
Root analytic conductor: \(1.70953\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{366} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 366,\ (\ :1/2),\ 0.977 - 0.210i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55043 + 0.164748i\)
\(L(\frac12)\) \(\approx\) \(1.55043 + 0.164748i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.777 + 0.629i)T \)
3 \( 1 + (-1.66 - 0.480i)T \)
61 \( 1 + (-7.74 + 1.03i)T \)
good5 \( 1 + (-0.665 - 0.739i)T + (-0.522 + 4.97i)T^{2} \)
7 \( 1 + (-2.47 - 0.949i)T + (5.20 + 4.68i)T^{2} \)
11 \( 1 + (2.73 + 2.73i)T + 11iT^{2} \)
13 \( 1 + (1.80 - 3.13i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.77 - 1.80i)T + (6.91 + 15.5i)T^{2} \)
19 \( 1 + (0.198 - 0.444i)T + (-12.7 - 14.1i)T^{2} \)
23 \( 1 + (0.245 + 0.482i)T + (-13.5 + 18.6i)T^{2} \)
29 \( 1 + (0.565 - 2.10i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (1.08 + 1.34i)T + (-6.44 + 30.3i)T^{2} \)
37 \( 1 + (-4.97 + 0.787i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (7.42 + 5.39i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-6.93 + 4.50i)T + (17.4 - 39.2i)T^{2} \)
47 \( 1 + (0.961 - 0.554i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.55 + 1.81i)T + (31.1 + 42.8i)T^{2} \)
59 \( 1 + (-1.41 + 1.74i)T + (-12.2 - 57.7i)T^{2} \)
67 \( 1 + (-2.76 - 0.144i)T + (66.6 + 7.00i)T^{2} \)
71 \( 1 + (1.15 - 0.0604i)T + (70.6 - 7.42i)T^{2} \)
73 \( 1 + (6.85 - 7.61i)T + (-7.63 - 72.6i)T^{2} \)
79 \( 1 + (6.98 + 10.7i)T + (-32.1 + 72.1i)T^{2} \)
83 \( 1 + (9.60 - 1.00i)T + (81.1 - 17.2i)T^{2} \)
89 \( 1 + (13.2 + 2.09i)T + (84.6 + 27.5i)T^{2} \)
97 \( 1 + (18.0 + 1.89i)T + (94.8 + 20.1i)T^{2} \)
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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

−11.20478194044586514467893556002, −10.42535561002639678248742599562, −9.638185464156148711028058642630, −8.588434057315404600266361562064, −8.094043252893886479019863998289, −7.05149217113254709630938082527, −5.47857158417581293480773825930, −4.18185825641838296123457181093, −2.85449535018262991509533097126, −1.87277103164742816396755649795, 1.40373039912616216690807206107, 2.76727502992408172367416106590, 4.55484303817782993562378426878, 5.50474446049863434202314689155, 7.11200312720341722496267887518, 7.75297226160117704374861300761, 8.352275223021008525414340990237, 9.576080224396073855873775672130, 10.05450461727525154726848405128, 11.21033220925022816058226165475

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