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Properties

Label 2-2268-7.4-c1-0-10
Degree $2$
Conductor $2268$
Sign $0.844 - 0.535i$
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  + (−0.515 − 0.892i)5-s + (1.55 + 2.14i)7-s + (0.792 − 1.37i)11-s + 5.04·13-s + (−2.58 + 4.47i)17-s + (−0.392 − 0.680i)19-s + (2.93 + 5.07i)23-s + (1.96 − 3.40i)25-s − 8.89·29-s + (0.575 − 0.996i)31-s + (1.10 − 2.49i)35-s + (4.07 + 7.06i)37-s + 7.74·41-s − 2.53·43-s + (−4.24 − 7.35i)47-s + ⋯
L(s)  = 1  + (−0.230 − 0.399i)5-s + (0.588 + 0.808i)7-s + (0.239 − 0.414i)11-s + 1.40·13-s + (−0.626 + 1.08i)17-s + (−0.0901 − 0.156i)19-s + (0.611 + 1.05i)23-s + (0.393 − 0.681i)25-s − 1.65·29-s + (0.103 − 0.179i)31-s + (0.187 − 0.421i)35-s + (0.670 + 1.16i)37-s + 1.20·41-s − 0.386·43-s + (−0.619 − 1.07i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 - 0.535i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.844 - 0.535i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (1621, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ 0.844 - 0.535i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.908003036\)
\(L(\frac12)\) \(\approx\) \(1.908003036\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.55 - 2.14i)T \)
good5 \( 1 + (0.515 + 0.892i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.792 + 1.37i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.04T + 13T^{2} \)
17 \( 1 + (2.58 - 4.47i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.392 + 0.680i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.93 - 5.07i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 8.89T + 29T^{2} \)
31 \( 1 + (-0.575 + 0.996i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.07 - 7.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.74T + 41T^{2} \)
43 \( 1 + 2.53T + 43T^{2} \)
47 \( 1 + (4.24 + 7.35i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.41 + 4.17i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.93 - 3.35i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.82 - 8.35i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.837 + 1.45i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 14.2T + 71T^{2} \)
73 \( 1 + (-3.04 + 5.27i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.15 - 7.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 14.2T + 83T^{2} \)
89 \( 1 + (-6.69 - 11.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.34T + 97T^{2} \)
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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

−8.846093364212180441947132796297, −8.473270095637424771660619298311, −7.79030617914873383745525652228, −6.60972383627698826105933698390, −5.92765999176495396917769713924, −5.21100928562871021278631020925, −4.18498669092835083867123276793, −3.44842166426036817480466246824, −2.14072803585353188156902097207, −1.13392656476277885499321753616, 0.793988881151274357379651167635, 2.01132677962845488735823373440, 3.26060416373022423863536772170, 4.09395327723802555495213896732, 4.81367037133698957687980373500, 5.87646114243807378160216137169, 6.80466633345931104526255415261, 7.33201142506870573745054120130, 8.107906159227119558672753023284, 8.995907149899419871963883868335

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