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Properties

Label 2-2268-84.83-c0-0-0
Degree $2$
Conductor $2268$
Sign $-0.5 - 0.866i$
Analytic cond. $1.13187$
Root an. cond. $1.06389$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + i·7-s + (0.707 + 0.707i)8-s − 1.93·11-s + (0.965 − 0.258i)14-s + (0.500 − 0.866i)16-s + (0.499 + 1.86i)22-s − 1.41·23-s − 25-s + (−0.499 − 0.866i)28-s − 1.41i·29-s + (−0.965 − 0.258i)32-s − 1.73·37-s + i·43-s + (1.67 − 0.965i)44-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + i·7-s + (0.707 + 0.707i)8-s − 1.93·11-s + (0.965 − 0.258i)14-s + (0.500 − 0.866i)16-s + (0.499 + 1.86i)22-s − 1.41·23-s − 25-s + (−0.499 − 0.866i)28-s − 1.41i·29-s + (−0.965 − 0.258i)32-s − 1.73·37-s + i·43-s + (1.67 − 0.965i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.5 - 0.866i$
Analytic conductor: \(1.13187\)
Root analytic conductor: \(1.06389\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (2267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :0),\ -0.5 - 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1293037995\)
\(L(\frac12)\) \(\approx\) \(0.1293037995\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
3 \( 1 \)
7 \( 1 - iT \)
good5 \( 1 + T^{2} \)
11 \( 1 + 1.93T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + 1.41T + T^{2} \)
29 \( 1 + 1.41iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + 1.73T + T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 0.517iT - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 1.73iT - T^{2} \)
71 \( 1 + 0.517T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - 1.73iT - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.724002243751421214064480683594, −8.706507986149249765616403691211, −8.045406208409450348546692878023, −7.62074507826101036530753349011, −6.09284538447056728053132239334, −5.40987901907198378454911371492, −4.63104160388952935719357980182, −3.53087138634598798226318466656, −2.54571677546383113448025812981, −1.98180303128715666715018780887, 0.089884613624563516795317136231, 1.83352631270095586773148056088, 3.35644560145126480182700926821, 4.28012973920979572385386216451, 5.19750255192702848350892666634, 5.74300247876116575078356033716, 6.84905891386181562034741900318, 7.43868963402416477390602494837, 8.043185358790295006517620938844, 8.680480376726208561488809243800

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