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Properties

Label 2-2268-252.83-c0-0-1
Degree $2$
Conductor $2268$
Sign $-0.342 - 0.939i$
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  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)7-s + 0.999·8-s − 0.999·10-s + (0.5 − 0.866i)11-s + (−0.499 − 0.866i)14-s + (−0.5 + 0.866i)16-s + 2·17-s − 19-s + (0.499 − 0.866i)20-s + (0.499 + 0.866i)22-s + (0.5 + 0.866i)23-s + 0.999·28-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)7-s + 0.999·8-s − 0.999·10-s + (0.5 − 0.866i)11-s + (−0.499 − 0.866i)14-s + (−0.5 + 0.866i)16-s + 2·17-s − 19-s + (0.499 − 0.866i)20-s + (0.499 + 0.866i)22-s + (0.5 + 0.866i)23-s + 0.999·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.342 - 0.939i$
Analytic conductor: \(1.13187\)
Root analytic conductor: \(1.06389\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (755, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :0),\ -0.342 - 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9532034267\)
\(L(\frac12)\) \(\approx\) \(0.9532034267\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 - 2T + T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−9.330731767870922871391990359605, −8.668203685649361174554344526751, −7.942961862813980020725325138650, −7.00349184768562797444175145588, −6.31045857831853096393718015722, −5.82087133313768225004815306785, −5.08151988694555214684390185138, −3.62836949148162856838261966575, −2.79042114980585679703878827036, −1.39237304901784740902156665551, 0.900733033996507382366563671355, 1.84036166752756837735137032785, 3.08800378580353510545928489736, 4.05181028342727094537079830054, 4.69331975974801751823377078366, 5.72889755239896148814197445517, 6.88252004702946130158920918209, 7.55357827734670214404778362723, 8.436168927982217944850746970470, 9.129467668550954652938644111116

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