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Properties

Label 2-2268-63.58-c1-0-3
Degree $2$
Conductor $2268$
Sign $-0.999 - 0.0354i$
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  + (2.00 + 3.47i)5-s + (−1.89 + 1.84i)7-s + (−0.885 + 1.53i)11-s + (−0.114 + 0.198i)13-s + (−3.04 − 5.27i)17-s + (−3.27 + 5.67i)19-s + (−0.769 − 1.33i)23-s + (−5.55 + 9.62i)25-s + (−0.271 − 0.469i)29-s + 4.55·31-s + (−10.2 − 2.86i)35-s + (1.54 − 2.66i)37-s + (−4.43 + 7.69i)41-s + (−2.12 − 3.67i)43-s + 0.757·47-s + ⋯
L(s)  = 1  + (0.897 + 1.55i)5-s + (−0.715 + 0.698i)7-s + (−0.267 + 0.462i)11-s + (−0.0317 + 0.0549i)13-s + (−0.739 − 1.28i)17-s + (−0.752 + 1.30i)19-s + (−0.160 − 0.277i)23-s + (−1.11 + 1.92i)25-s + (−0.0503 − 0.0872i)29-s + 0.817·31-s + (−1.72 − 0.484i)35-s + (0.253 − 0.438i)37-s + (−0.693 + 1.20i)41-s + (−0.323 − 0.560i)43-s + 0.110·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0354i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0354i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.999 - 0.0354i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (2053, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ -0.999 - 0.0354i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.034684821\)
\(L(\frac12)\) \(\approx\) \(1.034684821\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.89 - 1.84i)T \)
good5 \( 1 + (-2.00 - 3.47i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.885 - 1.53i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.114 - 0.198i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.04 + 5.27i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.27 - 5.67i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.769 + 1.33i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.271 + 0.469i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.55T + 31T^{2} \)
37 \( 1 + (-1.54 + 2.66i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.43 - 7.69i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.12 + 3.67i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 0.757T + 47T^{2} \)
53 \( 1 + (-3.19 - 5.53i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 5.17T + 59T^{2} \)
61 \( 1 - 12.5T + 61T^{2} \)
67 \( 1 + 6.18T + 67T^{2} \)
71 \( 1 + 13.9T + 71T^{2} \)
73 \( 1 + (5.08 + 8.81i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 11.5T + 79T^{2} \)
83 \( 1 + (8.66 + 15.0i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.04 + 8.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.91 - 8.50i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.537431010332267611344406954885, −8.805385013206207004828306677736, −7.68860426890265252349443321797, −6.91058281990703414079084017088, −6.29910510202743065376908650728, −5.77861894963211936752437150258, −4.65489075002833366692432795223, −3.39982896216240104976363968788, −2.62486321744597541476811208730, −2.00962439423830129909738996389, 0.33146591072060909785923568187, 1.47010181586565327534541415974, 2.58511098859180803642265077325, 3.93184633648215475422104021303, 4.61279266546663925752076873881, 5.47909738363850143033474240565, 6.25220958539515352665713721389, 6.91443636948477256166233312394, 8.194527990079482367235258858846, 8.655364238405395823609337450725

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