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Properties

Label 2-2268-63.41-c1-0-27
Degree $2$
Conductor $2268$
Sign $-0.873 + 0.486i$
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  + (−1.5 + 2.59i)5-s + (0.5 − 2.59i)7-s + (4.5 − 2.59i)11-s + (−3 − 1.73i)13-s − 6·17-s + 1.73i·19-s + (4.5 + 2.59i)23-s + (−2 − 3.46i)25-s + (−9 + 5.19i)29-s + (−4.5 − 2.59i)31-s + (6 + 5.19i)35-s + 37-s + (1.5 − 2.59i)41-s + (−5 − 8.66i)43-s + (3 + 5.19i)47-s + ⋯
L(s)  = 1  + (−0.670 + 1.16i)5-s + (0.188 − 0.981i)7-s + (1.35 − 0.783i)11-s + (−0.832 − 0.480i)13-s − 1.45·17-s + 0.397i·19-s + (0.938 + 0.541i)23-s + (−0.400 − 0.692i)25-s + (−1.67 + 0.964i)29-s + (−0.808 − 0.466i)31-s + (1.01 + 0.878i)35-s + 0.164·37-s + (0.234 − 0.405i)41-s + (−0.762 − 1.32i)43-s + (0.437 + 0.757i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.873 + 0.486i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.873 + 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.873 + 0.486i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ -0.873 + 0.486i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2818585877\)
\(L(\frac12)\) \(\approx\) \(0.2818585877\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 2.59i)T \)
good5 \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.5 + 2.59i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3 + 1.73i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 - 1.73iT - 19T^{2} \)
23 \( 1 + (-4.5 - 2.59i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (9 - 5.19i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.5 + 2.59i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - T + 37T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5 + 8.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (12 - 6.92i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.19iT - 71T^{2} \)
73 \( 1 + 3.46iT - 73T^{2} \)
79 \( 1 + (7 + 12.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3 + 5.19i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 9T + 89T^{2} \)
97 \( 1 + (6 - 3.46i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.818451898604271724250121376354, −7.54348912736802206381281372481, −7.28112402612049339777621314901, −6.59972199798844634021061682922, −5.64153391694718872080805739347, −4.42165660149710873602985285924, −3.70959219693282119338613136165, −3.06703515336781640809235490266, −1.65608631205647557733701236903, −0.095014008933280367276803188611, 1.52848527643472662795502894987, 2.43863365954322535998482304840, 3.91210225258567871081042198368, 4.61627555318613527602423536084, 5.10270497863213603869969747602, 6.32241608189410673258078795057, 7.01060814578107434101596984511, 7.905195555480777002153142779122, 8.825429849980032998087100299946, 9.161174751678670913217706957398

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