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Properties

Label 2-2268-21.5-c1-0-14
Degree $2$
Conductor $2268$
Sign $-0.184 - 0.982i$
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  + (−1.95 + 3.39i)5-s + (1.96 + 1.77i)7-s + (3.19 − 1.84i)11-s − 0.554i·13-s + (2.91 + 5.05i)17-s + (4.62 + 2.66i)19-s + (1.96 + 1.13i)23-s + (−5.16 − 8.94i)25-s + 4.08i·29-s + (7.00 − 4.04i)31-s + (−9.85 + 3.18i)35-s + (3.89 − 6.75i)37-s − 7.18·41-s + 1.50·43-s + (1.41 − 2.44i)47-s + ⋯
L(s)  = 1  + (−0.875 + 1.51i)5-s + (0.742 + 0.670i)7-s + (0.964 − 0.556i)11-s − 0.153i·13-s + (0.707 + 1.22i)17-s + (1.06 + 0.612i)19-s + (0.410 + 0.237i)23-s + (−1.03 − 1.78i)25-s + 0.758i·29-s + (1.25 − 0.726i)31-s + (−1.66 + 0.538i)35-s + (0.640 − 1.11i)37-s − 1.12·41-s + 0.229·43-s + (0.206 − 0.357i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.184 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.184 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.184 - 0.982i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (2105, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ -0.184 - 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.808409419\)
\(L(\frac12)\) \(\approx\) \(1.808409419\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.96 - 1.77i)T \)
good5 \( 1 + (1.95 - 3.39i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.19 + 1.84i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.554iT - 13T^{2} \)
17 \( 1 + (-2.91 - 5.05i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.62 - 2.66i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.96 - 1.13i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.08iT - 29T^{2} \)
31 \( 1 + (-7.00 + 4.04i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.89 + 6.75i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 7.18T + 41T^{2} \)
43 \( 1 - 1.50T + 43T^{2} \)
47 \( 1 + (-1.41 + 2.44i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.0415 - 0.0239i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.45 - 7.71i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.03 + 3.48i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.587 + 1.01i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.71iT - 71T^{2} \)
73 \( 1 + (3.52 - 2.03i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.97 + 3.41i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.69T + 83T^{2} \)
89 \( 1 + (2.71 - 4.69i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 16.1iT - 97T^{2} \)
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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

−9.165141930432984236766927036613, −8.246165963720707571792679606308, −7.78224617755438050030829370652, −6.96378701108215867307356597649, −6.14257981259873991138801071676, −5.47669255570930102332612229102, −4.13005992486789841520316920265, −3.49202229973856933187096576759, −2.67888033484685182805190111929, −1.33798265588698071817172361597, 0.76374315374517573361042705057, 1.40132108819806347676460463265, 3.10636546998227733410853476650, 4.21689340429718345809085402125, 4.72549783086934700340233887456, 5.26086117880841553224259863447, 6.67094762417171455055263342906, 7.44111108140560863648075285749, 8.017164263520897113994534872064, 8.771677073680540372159307545973

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