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Properties

Label 2-2268-63.4-c1-0-10
Degree $2$
Conductor $2268$
Sign $0.678 - 0.734i$
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  − 3·5-s + (2.5 + 0.866i)7-s + 3·11-s + (−1 + 1.73i)13-s + (1.5 − 2.59i)17-s + (0.5 + 0.866i)19-s − 3·23-s + 4·25-s + (−3 − 5.19i)29-s + (3.5 + 6.06i)31-s + (−7.5 − 2.59i)35-s + (0.5 + 0.866i)37-s + (3 − 5.19i)41-s + (2 + 3.46i)43-s + (−4.5 + 7.79i)47-s + ⋯
L(s)  = 1  − 1.34·5-s + (0.944 + 0.327i)7-s + 0.904·11-s + (−0.277 + 0.480i)13-s + (0.363 − 0.630i)17-s + (0.114 + 0.198i)19-s − 0.625·23-s + 0.800·25-s + (−0.557 − 0.964i)29-s + (0.628 + 1.08i)31-s + (−1.26 − 0.439i)35-s + (0.0821 + 0.142i)37-s + (0.468 − 0.811i)41-s + (0.304 + 0.528i)43-s + (−0.656 + 1.13i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.678 - 0.734i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ 0.678 - 0.734i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.453639509\)
\(L(\frac12)\) \(\approx\) \(1.453639509\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 0.866i)T \)
good5 \( 1 + 3T + 5T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 3T + 23T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.5 - 6.06i)T + (-15.5 + 26.8i)T^{2} \)
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 \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.5 - 7.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.5 - 7.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.5 + 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6 - 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.5 - 12.9i)T + (-44.5 + 77.0i)T^{2} \)
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

−9.026861867605260381224244182765, −8.208077386255098953944052598795, −7.70688963258776613259040676865, −6.98277802225329297155758503026, −6.01135986598653580852330347102, −4.94948962517530743792586606247, −4.26766695122424809613125007561, −3.55653432205470019691054163608, −2.30405514557169386913691066221, −1.02007408875464759529332644698, 0.64228351253592272391219834932, 1.90531808697282450301918967626, 3.38290175680366818695727836974, 4.02777428217235854159078374705, 4.73478362570569390012341664446, 5.72195888197127843599900900255, 6.77348592691730488382292906001, 7.57573961418871796428557587065, 8.046775282463125525462751855021, 8.686954904080750992429404424793

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