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Properties

Label 2-2268-63.25-c1-0-25
Degree $2$
Conductor $2268$
Sign $0.246 + 0.969i$
Analytic cond. $18.1100$
Root an. cond. $4.25559$
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  + (2.14 − 3.71i)5-s + (1.23 + 2.33i)7-s + (1.90 + 3.30i)11-s + (−1.64 − 2.84i)13-s + (0.405 − 0.702i)17-s + (−3.54 − 6.14i)19-s + (3.23 − 5.60i)23-s + (−6.69 − 11.5i)25-s + (−1.90 + 3.30i)29-s + 3.28·31-s + (11.3 + 0.413i)35-s + (2.88 + 4.99i)37-s + (−1.04 − 1.81i)41-s + (4.38 − 7.59i)43-s + 3.33·47-s + ⋯
L(s)  = 1  + (0.958 − 1.66i)5-s + (0.468 + 0.883i)7-s + (0.574 + 0.995i)11-s + (−0.455 − 0.789i)13-s + (0.0983 − 0.170i)17-s + (−0.814 − 1.41i)19-s + (0.675 − 1.16i)23-s + (−1.33 − 2.31i)25-s + (−0.353 + 0.612i)29-s + 0.590·31-s + (1.91 + 0.0698i)35-s + (0.473 + 0.820i)37-s + (−0.163 − 0.283i)41-s + (0.668 − 1.15i)43-s + 0.486·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.246 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.246 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.246 + 0.969i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ 0.246 + 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.195175470\)
\(L(\frac12)\) \(\approx\) \(2.195175470\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-1.23 - 2.33i)T \)
good5 \( 1 + (-2.14 + 3.71i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.90 - 3.30i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.64 + 2.84i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.405 + 0.702i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.54 + 6.14i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.23 + 5.60i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.90 - 3.30i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.28T + 31T^{2} \)
37 \( 1 + (-2.88 - 4.99i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.04 + 1.81i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.38 + 7.59i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.33T + 47T^{2} \)
53 \( 1 + (4.93 - 8.54i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 3.47T + 59T^{2} \)
61 \( 1 - 5.95T + 61T^{2} \)
67 \( 1 + 3.52T + 67T^{2} \)
71 \( 1 - 6.05T + 71T^{2} \)
73 \( 1 + (-5.19 + 8.99i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 5.14T + 79T^{2} \)
83 \( 1 + (-0.856 + 1.48i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.26 + 10.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.522 - 0.905i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.791784564347837773449953760419, −8.500227371341571988519975209964, −7.34881009121483367150737487985, −6.37010013417831448680697352346, −5.54417823245155498014348539387, −4.78487923530294643843620116847, −4.51199901889230226580709880018, −2.65633771330871286550296737677, −1.93232395216013555224509255150, −0.78034174689316619182903865738, 1.42623701856950031719874630061, 2.36914728409461176620708382199, 3.47508231035688522544968625966, 4.08249630158622048833314840720, 5.47707661327782888981030096548, 6.20612145145707275879959971273, 6.76876731174464847948408795173, 7.51578007919030155930415139164, 8.282336611049655857191287838134, 9.526179971588430876296480063675

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