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Properties

Label 2-366-183.104-c1-0-18
Degree $2$
Conductor $366$
Sign $-0.964 - 0.264i$
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 + (0.899 − 1.48i)3-s + (0.207 + 0.978i)4-s + (−2.42 − 2.68i)5-s + (−1.63 + 0.584i)6-s + (−0.882 − 0.338i)7-s + (0.453 − 0.891i)8-s + (−1.38 − 2.66i)9-s + (0.189 + 3.61i)10-s + (0.277 + 0.277i)11-s + (1.63 + 0.571i)12-s + (−2.31 + 4.01i)13-s + (0.472 + 0.818i)14-s + (−6.15 + 1.16i)15-s + (−0.913 + 0.406i)16-s + (0.949 + 0.616i)17-s + ⋯
L(s)  = 1  + (−0.549 − 0.444i)2-s + (0.519 − 0.854i)3-s + (0.103 + 0.489i)4-s + (−1.08 − 1.20i)5-s + (−0.665 + 0.238i)6-s + (−0.333 − 0.128i)7-s + (0.160 − 0.315i)8-s + (−0.460 − 0.887i)9-s + (0.0598 + 1.14i)10-s + (0.0836 + 0.0836i)11-s + (0.471 + 0.165i)12-s + (−0.642 + 1.11i)13-s + (0.126 + 0.218i)14-s + (−1.59 + 0.301i)15-s + (−0.228 + 0.101i)16-s + (0.230 + 0.149i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 - 0.264i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 - 0.264i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(366\)    =    \(2 \cdot 3 \cdot 61\)
Sign: $-0.964 - 0.264i$
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: \((2,\ 366,\ (\ :1/2),\ -0.964 - 0.264i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0827236 + 0.614409i\)
\(L(\frac12)\) \(\approx\) \(0.0827236 + 0.614409i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.899 + 1.48i)T \)
61 \( 1 + (-0.637 + 7.78i)T \)
good5 \( 1 + (2.42 + 2.68i)T + (-0.522 + 4.97i)T^{2} \)
7 \( 1 + (0.882 + 0.338i)T + (5.20 + 4.68i)T^{2} \)
11 \( 1 + (-0.277 - 0.277i)T + 11iT^{2} \)
13 \( 1 + (2.31 - 4.01i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.949 - 0.616i)T + (6.91 + 15.5i)T^{2} \)
19 \( 1 + (-0.368 + 0.827i)T + (-12.7 - 14.1i)T^{2} \)
23 \( 1 + (1.93 + 3.80i)T + (-13.5 + 18.6i)T^{2} \)
29 \( 1 + (-0.401 + 1.49i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (3.76 + 4.65i)T + (-6.44 + 30.3i)T^{2} \)
37 \( 1 + (-7.60 + 1.20i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (2.09 + 1.52i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-2.80 + 1.82i)T + (17.4 - 39.2i)T^{2} \)
47 \( 1 + (-0.438 + 0.253i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (12.3 + 6.31i)T + (31.1 + 42.8i)T^{2} \)
59 \( 1 + (7.14 - 8.81i)T + (-12.2 - 57.7i)T^{2} \)
67 \( 1 + (-10.7 - 0.561i)T + (66.6 + 7.00i)T^{2} \)
71 \( 1 + (4.21 - 0.220i)T + (70.6 - 7.42i)T^{2} \)
73 \( 1 + (-9.89 + 10.9i)T + (-7.63 - 72.6i)T^{2} \)
79 \( 1 + (1.55 + 2.39i)T + (-32.1 + 72.1i)T^{2} \)
83 \( 1 + (-16.9 + 1.77i)T + (81.1 - 17.2i)T^{2} \)
89 \( 1 + (-11.0 - 1.74i)T + (84.6 + 27.5i)T^{2} \)
97 \( 1 + (-1.31 - 0.138i)T + (94.8 + 20.1i)T^{2} \)
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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

−11.21274645775660755883495846054, −9.621972685151668118285467894622, −9.047246372491971436826199866145, −8.109287340114890652488530352636, −7.52185047451158599883209061852, −6.41623017594788628685609749093, −4.62710543260838321596316940418, −3.61106394130459300141627792438, −2.02234139643135876109203479052, −0.46051327598609432315430855189, 2.81060276278353756953547279920, 3.62804229188270554158962907471, 5.05204779402458907275955175341, 6.33563266960703135119447026333, 7.65674755051988230964329611906, 7.890370266774327918639536919788, 9.231844789662391405871642814234, 10.04964992559507258124134770839, 10.78834458492827736634985763004, 11.50513467630456515771819871471

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