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Properties

Label 2-366-61.37-c2-0-11
Degree $2$
Conductor $366$
Sign $0.977 - 0.210i$
Analytic cond. $9.97277$
Root an. cond. $3.15797$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.642 + 1.26i)2-s + (1.64 − 0.535i)3-s + (−1.17 − 1.61i)4-s + (−1.15 − 1.58i)5-s + (−0.383 + 2.41i)6-s + (−6.58 + 3.35i)7-s + (2.79 − 0.442i)8-s + (2.42 − 1.76i)9-s + (2.73 − 0.433i)10-s + (4.59 − 4.59i)11-s + (−2.80 − 2.03i)12-s + 14.4·13-s − 10.4i·14-s + (−2.74 − 1.99i)15-s + (−1.23 + 3.80i)16-s + (−2.16 + 13.6i)17-s + ⋯
L(s)  = 1  + (−0.321 + 0.630i)2-s + (0.549 − 0.178i)3-s + (−0.293 − 0.404i)4-s + (−0.230 − 0.316i)5-s + (−0.0638 + 0.403i)6-s + (−0.941 + 0.479i)7-s + (0.349 − 0.0553i)8-s + (0.269 − 0.195i)9-s + (0.273 − 0.0433i)10-s + (0.417 − 0.417i)11-s + (−0.233 − 0.169i)12-s + 1.11·13-s − 0.746i·14-s + (−0.182 − 0.132i)15-s + (−0.0772 + 0.237i)16-s + (−0.127 + 0.803i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.210i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 366 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.977 - 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(366\)    =    \(2 \cdot 3 \cdot 61\)
Sign: $0.977 - 0.210i$
Analytic conductor: \(9.97277\)
Root analytic conductor: \(3.15797\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{366} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 366,\ (\ :1),\ 0.977 - 0.210i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.56702 + 0.166679i\)
\(L(\frac12)\) \(\approx\) \(1.56702 + 0.166679i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.642 - 1.26i)T \)
3 \( 1 + (-1.64 + 0.535i)T \)
61 \( 1 + (59.2 - 14.3i)T \)
good5 \( 1 + (1.15 + 1.58i)T + (-7.72 + 23.7i)T^{2} \)
7 \( 1 + (6.58 - 3.35i)T + (28.8 - 39.6i)T^{2} \)
11 \( 1 + (-4.59 + 4.59i)T - 121iT^{2} \)
13 \( 1 - 14.4T + 169T^{2} \)
17 \( 1 + (2.16 - 13.6i)T + (-274. - 89.3i)T^{2} \)
19 \( 1 + (-20.4 + 6.65i)T + (292. - 212. i)T^{2} \)
23 \( 1 + (-38.9 - 6.17i)T + (503. + 163. i)T^{2} \)
29 \( 1 + (-23.9 + 23.9i)T - 841iT^{2} \)
31 \( 1 + (12.1 + 23.8i)T + (-564. + 777. i)T^{2} \)
37 \( 1 + (8.49 + 16.6i)T + (-804. + 1.10e3i)T^{2} \)
41 \( 1 + (-3.25 - 1.05i)T + (1.35e3 + 988. i)T^{2} \)
43 \( 1 + (34.3 - 5.44i)T + (1.75e3 - 571. i)T^{2} \)
47 \( 1 - 81.9T + 2.20e3T^{2} \)
53 \( 1 + (-4.55 - 28.7i)T + (-2.67e3 + 868. i)T^{2} \)
59 \( 1 + (-60.8 - 31.0i)T + (2.04e3 + 2.81e3i)T^{2} \)
67 \( 1 + (10.8 - 68.6i)T + (-4.26e3 - 1.38e3i)T^{2} \)
71 \( 1 + (83.2 - 13.1i)T + (4.79e3 - 1.55e3i)T^{2} \)
73 \( 1 + (2.53 + 1.83i)T + (1.64e3 + 5.06e3i)T^{2} \)
79 \( 1 + (8.87 + 56.0i)T + (-5.93e3 + 1.92e3i)T^{2} \)
83 \( 1 + (-22.8 - 70.1i)T + (-5.57e3 + 4.04e3i)T^{2} \)
89 \( 1 + (60.5 + 30.8i)T + (4.65e3 + 6.40e3i)T^{2} \)
97 \( 1 + (-99.5 - 32.3i)T + (7.61e3 + 5.53e3i)T^{2} \)
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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

−11.16929547235661157093817468566, −10.03798554881546405073587217475, −8.964346630632287465476248057548, −8.698924924471854354583443632206, −7.49786334239069267974067377042, −6.48605398523595334308575839289, −5.68334401327266611974662393031, −4.13715180985805996044416141709, −2.95729068928855279226316932339, −0.987068408175095630782964210382, 1.16883225630282471944483676805, 3.05626434348628470275789022438, 3.58373323854443078359589436836, 5.00058149327997470897483414998, 6.73672990715956223072547966483, 7.36271056568865086847222145307, 8.739046891207607380619741052161, 9.312049750117952794541777089011, 10.26669612311219706216299133936, 10.99578700535276903020687895955

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