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Properties

Label 2-1116-124.87-c0-0-0
Degree $2$
Conductor $1116$
Sign $0.602 - 0.798i$
Analytic cond. $0.556956$
Root an. cond. $0.746295$
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  + i·2-s − 4-s + (0.5 − 0.866i)5-s + (−0.866 + 0.5i)7-s − i·8-s + (0.866 + 0.5i)10-s + (0.866 + 0.5i)11-s + (0.5 − 0.866i)13-s + (−0.5 − 0.866i)14-s + 16-s + (0.5 + 0.866i)17-s + (0.866 − 0.5i)19-s + (−0.5 + 0.866i)20-s + (−0.5 + 0.866i)22-s + (0.866 + 0.5i)26-s + ⋯
L(s)  = 1  + i·2-s − 4-s + (0.5 − 0.866i)5-s + (−0.866 + 0.5i)7-s − i·8-s + (0.866 + 0.5i)10-s + (0.866 + 0.5i)11-s + (0.5 − 0.866i)13-s + (−0.5 − 0.866i)14-s + 16-s + (0.5 + 0.866i)17-s + (0.866 − 0.5i)19-s + (−0.5 + 0.866i)20-s + (−0.5 + 0.866i)22-s + (0.866 + 0.5i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.602 - 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.602 - 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1116\)    =    \(2^{2} \cdot 3^{2} \cdot 31\)
Sign: $0.602 - 0.798i$
Analytic conductor: \(0.556956\)
Root analytic conductor: \(0.746295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1116} (955, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1116,\ (\ :0),\ 0.602 - 0.798i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.015223097\)
\(L(\frac12)\) \(\approx\) \(1.015223097\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.797436186750969840497346921837, −9.102194323487606699406571631465, −8.716393248804543117042353037700, −7.61395621670180171237510305135, −6.75738431060650757701392096164, −5.79741031208453825217921125763, −5.41133926647758289575073481390, −4.21543338030380175554953567885, −3.20521755991001567859526013285, −1.27229463794011922222899848491, 1.29610807322623552332265508911, 2.75331563927132718503450088724, 3.46644045804636655364032790428, 4.35329586426174275798526098259, 5.73902641479683288670924875470, 6.47785703853219884531475605215, 7.37741019449462285087211132918, 8.560721566302760111996929986738, 9.585590822026219929534396316567, 9.780928082194643669998637910793

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