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Properties

Label 2-2268-63.58-c1-0-25
Degree $2$
Conductor $2268$
Sign $0.415 + 0.909i$
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 + 1.73i)5-s + (2 − 1.73i)7-s + (−1 + 1.73i)11-s + (1.5 − 2.59i)13-s + (−4 − 6.92i)17-s + (0.5 − 0.866i)19-s + (−4 − 6.92i)23-s + (0.500 − 0.866i)25-s + (−2 − 3.46i)29-s + 3·31-s + (5 + 1.73i)35-s + (0.5 − 0.866i)37-s + (−3 + 5.19i)41-s + (−5.5 − 9.52i)43-s + 6·47-s + ⋯
L(s)  = 1  + (0.447 + 0.774i)5-s + (0.755 − 0.654i)7-s + (−0.301 + 0.522i)11-s + (0.416 − 0.720i)13-s + (−0.970 − 1.68i)17-s + (0.114 − 0.198i)19-s + (−0.834 − 1.44i)23-s + (0.100 − 0.173i)25-s + (−0.371 − 0.643i)29-s + 0.538·31-s + (0.845 + 0.292i)35-s + (0.0821 − 0.142i)37-s + (−0.468 + 0.811i)41-s + (−0.838 − 1.45i)43-s + 0.875·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.415 + 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.415 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.415 + 0.909i$
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: \((2,\ 2268,\ (\ :1/2),\ 0.415 + 0.909i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.771304709\)
\(L(\frac12)\) \(\approx\) \(1.771304709\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + 1.73i)T \)
good5 \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.5 + 2.59i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (4 + 6.92i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4 + 6.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2 + 3.46i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3T + 31T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.5 + 9.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 + (-6 - 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 - 13T + 67T^{2} \)
71 \( 1 + 10T + 71T^{2} \)
73 \( 1 + (-5.5 - 9.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 3T + 79T^{2} \)
83 \( 1 + (1 + 1.73i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5 + 8.66i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.793208803279708033121274967227, −8.088055200662331662843651367220, −7.19905446495384035248345791112, −6.74714563035558892810252706902, −5.74896696538666990325079089328, −4.81010935044856428897837040456, −4.13966417382138711149414154104, −2.80951950098308471228437779420, −2.17818439064586143987430669843, −0.60301918906794572201867742066, 1.46153644938947387268722094698, 2.02998938511766737556348125152, 3.51131856340625239343958472360, 4.39495063421745366945109745250, 5.33653311906833003067491435070, 5.85148063232732749351307855217, 6.70603804533987599323116807290, 7.907974488086105328692353059732, 8.485862272427510297256843424548, 8.981943095355619302911644950567

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