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Properties

Label 2-1380-1380.539-c0-0-0
Degree $2$
Conductor $1380$
Sign $-0.898 + 0.438i$
Analytic cond. $0.688709$
Root an. cond. $0.829885$
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  + (−0.755 + 0.654i)2-s + (0.281 + 0.959i)3-s + (0.142 − 0.989i)4-s + (−0.415 + 0.909i)5-s + (−0.841 − 0.540i)6-s + (−0.540 + 1.84i)7-s + (0.540 + 0.841i)8-s + (−0.841 + 0.540i)9-s + (−0.281 − 0.959i)10-s + (0.989 − 0.142i)12-s + (−0.797 − 1.74i)14-s + (−0.989 − 0.142i)15-s + (−0.959 − 0.281i)16-s + (0.281 − 0.959i)18-s + (0.841 + 0.540i)20-s − 1.91·21-s + ⋯
L(s)  = 1  + (−0.755 + 0.654i)2-s + (0.281 + 0.959i)3-s + (0.142 − 0.989i)4-s + (−0.415 + 0.909i)5-s + (−0.841 − 0.540i)6-s + (−0.540 + 1.84i)7-s + (0.540 + 0.841i)8-s + (−0.841 + 0.540i)9-s + (−0.281 − 0.959i)10-s + (0.989 − 0.142i)12-s + (−0.797 − 1.74i)14-s + (−0.989 − 0.142i)15-s + (−0.959 − 0.281i)16-s + (0.281 − 0.959i)18-s + (0.841 + 0.540i)20-s − 1.91·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.898 + 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.898 + 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.898 + 0.438i$
Analytic conductor: \(0.688709\)
Root analytic conductor: \(0.829885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (539, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :0),\ -0.898 + 0.438i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6085462383\)
\(L(\frac12)\) \(\approx\) \(0.6085462383\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.755 - 0.654i)T \)
3 \( 1 + (-0.281 - 0.959i)T \)
5 \( 1 + (0.415 - 0.909i)T \)
23 \( 1 + (-0.755 + 0.654i)T \)
good7 \( 1 + (0.540 - 1.84i)T + (-0.841 - 0.540i)T^{2} \)
11 \( 1 + (0.142 + 0.989i)T^{2} \)
13 \( 1 + (-0.841 + 0.540i)T^{2} \)
17 \( 1 + (0.959 - 0.281i)T^{2} \)
19 \( 1 + (-0.959 - 0.281i)T^{2} \)
29 \( 1 + (-1.80 + 0.258i)T + (0.959 - 0.281i)T^{2} \)
31 \( 1 + (-0.415 + 0.909i)T^{2} \)
37 \( 1 + (-0.654 + 0.755i)T^{2} \)
41 \( 1 + (0.512 + 0.234i)T + (0.654 + 0.755i)T^{2} \)
43 \( 1 + (-0.153 + 0.239i)T + (-0.415 - 0.909i)T^{2} \)
47 \( 1 - 0.830iT - T^{2} \)
53 \( 1 + (-0.841 - 0.540i)T^{2} \)
59 \( 1 + (0.841 - 0.540i)T^{2} \)
61 \( 1 + (-1.07 - 1.66i)T + (-0.415 + 0.909i)T^{2} \)
67 \( 1 + (1.27 - 1.10i)T + (0.142 - 0.989i)T^{2} \)
71 \( 1 + (-0.142 + 0.989i)T^{2} \)
73 \( 1 + (0.959 + 0.281i)T^{2} \)
79 \( 1 + (0.841 - 0.540i)T^{2} \)
83 \( 1 + (0.449 + 0.983i)T + (-0.654 + 0.755i)T^{2} \)
89 \( 1 + (-1.10 - 0.708i)T + (0.415 + 0.909i)T^{2} \)
97 \( 1 + (-0.654 - 0.755i)T^{2} \)
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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

−10.10295979080772331026150475611, −9.303548280228258143316380626007, −8.660068793666361508382488765144, −8.145936784961901135722592659402, −6.94595396057941310396876211435, −6.16778925835051956898555284535, −5.46780216558317798940274553702, −4.47320313050209808586952170450, −2.99034784139045130375492295521, −2.44332762015097796960382548994, 0.65488878432635551092815441478, 1.49904962779147860604624092949, 3.08622338534655516318594460510, 3.82438592875645222443977645697, 4.84421511518242418855244090887, 6.51432455698625909723726895626, 7.12663693144195432120669711603, 7.81959506112461435197677706756, 8.428857520212084354618801061467, 9.297782999973405018059647810281

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