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Properties

Label 2-2268-252.83-c0-0-14
Degree $2$
Conductor $2268$
Sign $-0.819 + 0.573i$
Analytic cond. $1.13187$
Root an. cond. $1.06389$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00i·4-s + (−0.866 − 0.5i)7-s + (−0.707 − 0.707i)8-s + (0.258 − 0.448i)11-s + (−0.965 + 0.258i)14-s − 1.00·16-s + (−0.133 − 0.5i)22-s + (−0.707 − 1.22i)23-s + (0.5 − 0.866i)25-s + (−0.500 + 0.866i)28-s + (−1.22 − 0.707i)29-s + (−0.707 + 0.707i)32-s + 1.73·37-s + (−0.866 − 0.5i)43-s + (−0.448 − 0.258i)44-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00i·4-s + (−0.866 − 0.5i)7-s + (−0.707 − 0.707i)8-s + (0.258 − 0.448i)11-s + (−0.965 + 0.258i)14-s − 1.00·16-s + (−0.133 − 0.5i)22-s + (−0.707 − 1.22i)23-s + (0.5 − 0.866i)25-s + (−0.500 + 0.866i)28-s + (−1.22 − 0.707i)29-s + (−0.707 + 0.707i)32-s + 1.73·37-s + (−0.866 − 0.5i)43-s + (−0.448 − 0.258i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.819 + 0.573i$
Analytic conductor: \(1.13187\)
Root analytic conductor: \(1.06389\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (755, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :0),\ -0.819 + 0.573i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.333288569\)
\(L(\frac12)\) \(\approx\) \(1.333288569\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 \)
7 \( 1 + (0.866 + 0.5i)T \)
good5 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.258 + 0.448i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - 1.73T + T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 - 1.93iT - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + 1.93T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−9.174275871944613109019411708588, −8.209559548024047881822390546272, −7.14194516488153600990798508474, −6.26655177072619160246298928309, −5.88525596545774329137725650550, −4.60385938540037747337465033492, −4.00675943593909873809539617207, −3.11028197782058620216342383859, −2.21865544244379298600872732754, −0.70514448429133486126446239676, 2.01023225416444099039705039666, 3.20060503547983149473298624320, 3.79423262523120809815961608873, 4.90346848693405910483496339464, 5.65025341426476765299008612945, 6.33370463914481244758299445050, 7.12092238489690454935771491005, 7.74211737067261562123063334609, 8.714442250763650158283281606724, 9.393503094259113710443814586401

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