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Properties

Label 2-366-183.101-c1-0-15
Degree $2$
Conductor $366$
Sign $0.619 + 0.784i$
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  + (−0.965 + 0.258i)2-s + (1.72 − 0.120i)3-s + (0.866 − 0.499i)4-s + (1.01 − 1.76i)5-s + (−1.63 + 0.563i)6-s + (−0.746 − 2.78i)7-s + (−0.707 + 0.707i)8-s + (2.97 − 0.417i)9-s + (−0.526 + 1.96i)10-s + (−2.79 + 2.79i)11-s + (1.43 − 0.968i)12-s + (3.45 − 5.97i)13-s + (1.44 + 2.49i)14-s + (1.54 − 3.16i)15-s + (0.500 − 0.866i)16-s + (−5.55 − 1.48i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.997 − 0.0697i)3-s + (0.433 − 0.249i)4-s + (0.454 − 0.787i)5-s + (−0.668 + 0.230i)6-s + (−0.282 − 1.05i)7-s + (−0.249 + 0.249i)8-s + (0.990 − 0.139i)9-s + (−0.166 + 0.621i)10-s + (−0.843 + 0.843i)11-s + (0.414 − 0.279i)12-s + (0.957 − 1.65i)13-s + (0.385 + 0.667i)14-s + (0.398 − 0.817i)15-s + (0.125 − 0.216i)16-s + (−1.34 − 0.361i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.619 + 0.784i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.619 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(366\)    =    \(2 \cdot 3 \cdot 61\)
Sign: $0.619 + 0.784i$
Analytic conductor: \(2.92252\)
Root analytic conductor: \(1.70953\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{366} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 366,\ (\ :1/2),\ 0.619 + 0.784i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25347 - 0.607223i\)
\(L(\frac12)\) \(\approx\) \(1.25347 - 0.607223i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 + (-1.72 + 0.120i)T \)
61 \( 1 + (1.93 - 7.56i)T \)
good5 \( 1 + (-1.01 + 1.76i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.746 + 2.78i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (2.79 - 2.79i)T - 11iT^{2} \)
13 \( 1 + (-3.45 + 5.97i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (5.55 + 1.48i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (3.21 - 1.85i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.37 - 3.37i)T + 23iT^{2} \)
29 \( 1 + (-3.70 - 0.991i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (1.19 - 4.47i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (0.887 + 0.887i)T + 37iT^{2} \)
41 \( 1 - 1.68T + 41T^{2} \)
43 \( 1 + (-11.7 + 3.14i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-3.94 + 2.27i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.0628 + 0.0628i)T + 53iT^{2} \)
59 \( 1 + (-2.69 - 10.0i)T + (-51.0 + 29.5i)T^{2} \)
67 \( 1 + (5.42 - 1.45i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (4.15 + 1.11i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-6.30 - 10.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.0557 + 0.207i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (5.72 + 3.30i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-8.47 + 8.47i)T - 89iT^{2} \)
97 \( 1 + (11.8 - 6.84i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.60693810023440517117749934315, −10.43615154588538063799775199093, −9.235246276861941080141181257536, −8.599749086526641004339047574489, −7.65246345282163603572735247309, −6.91911335641948111922148414142, −5.44720470970356691638159254710, −4.14916263964411238823151797811, −2.69064899160633818582597465017, −1.14871952340589510177196720835, 2.18019130114334176298829942048, 2.78168542512035677951142405938, 4.27382017547311627531824427372, 6.20707020142637281226603726713, 6.75179500800198266463600724031, 8.195547322128393911191315401597, 8.905679136593732373173402134266, 9.373828442961952779712842227381, 10.74336441269097479387723503039, 11.06647778434692123705651277446

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