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Properties

Label 2-1380-1380.1259-c0-0-2
Degree $2$
Conductor $1380$
Sign $0.991 - 0.129i$
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  + (−0.909 − 0.415i)2-s + (0.755 + 0.654i)3-s + (0.654 + 0.755i)4-s + (0.841 − 0.540i)5-s + (−0.415 − 0.909i)6-s + (−0.281 − 0.0405i)7-s + (−0.281 − 0.959i)8-s + (0.142 + 0.989i)9-s + (−0.989 + 0.142i)10-s + i·12-s + (0.239 + 0.153i)14-s + (0.989 + 0.142i)15-s + (−0.142 + 0.989i)16-s + (0.281 − 0.959i)18-s + (0.959 + 0.281i)20-s + (−0.186 − 0.215i)21-s + ⋯
L(s)  = 1  + (−0.909 − 0.415i)2-s + (0.755 + 0.654i)3-s + (0.654 + 0.755i)4-s + (0.841 − 0.540i)5-s + (−0.415 − 0.909i)6-s + (−0.281 − 0.0405i)7-s + (−0.281 − 0.959i)8-s + (0.142 + 0.989i)9-s + (−0.989 + 0.142i)10-s + i·12-s + (0.239 + 0.153i)14-s + (0.989 + 0.142i)15-s + (−0.142 + 0.989i)16-s + (0.281 − 0.959i)18-s + (0.959 + 0.281i)20-s + (−0.186 − 0.215i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.991 - 0.129i$
Analytic conductor: \(0.688709\)
Root analytic conductor: \(0.829885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (1259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1380,\ (\ :0),\ 0.991 - 0.129i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.060601799\)
\(L(\frac12)\) \(\approx\) \(1.060601799\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.909 + 0.415i)T \)
3 \( 1 + (-0.755 - 0.654i)T \)
5 \( 1 + (-0.841 + 0.540i)T \)
23 \( 1 + (-0.909 - 0.415i)T \)
good7 \( 1 + (0.281 + 0.0405i)T + (0.959 + 0.281i)T^{2} \)
11 \( 1 + (0.654 - 0.755i)T^{2} \)
13 \( 1 + (0.959 - 0.281i)T^{2} \)
17 \( 1 + (0.142 + 0.989i)T^{2} \)
19 \( 1 + (-0.142 + 0.989i)T^{2} \)
29 \( 1 + (-0.817 - 0.708i)T + (0.142 + 0.989i)T^{2} \)
31 \( 1 + (-0.841 + 0.540i)T^{2} \)
37 \( 1 + (0.415 + 0.909i)T^{2} \)
41 \( 1 + (1.07 + 1.66i)T + (-0.415 + 0.909i)T^{2} \)
43 \( 1 + (-0.368 + 1.25i)T + (-0.841 - 0.540i)T^{2} \)
47 \( 1 - 1.68iT - T^{2} \)
53 \( 1 + (0.959 + 0.281i)T^{2} \)
59 \( 1 + (-0.959 + 0.281i)T^{2} \)
61 \( 1 + (-0.425 - 1.45i)T + (-0.841 + 0.540i)T^{2} \)
67 \( 1 + (1.74 + 0.797i)T + (0.654 + 0.755i)T^{2} \)
71 \( 1 + (-0.654 - 0.755i)T^{2} \)
73 \( 1 + (0.142 - 0.989i)T^{2} \)
79 \( 1 + (-0.959 + 0.281i)T^{2} \)
83 \( 1 + (-0.474 - 0.304i)T + (0.415 + 0.909i)T^{2} \)
89 \( 1 + (0.797 + 0.234i)T + (0.841 + 0.540i)T^{2} \)
97 \( 1 + (0.415 - 0.909i)T^{2} \)
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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

−9.720356942883713729002989722262, −8.935746772553779150825129008722, −8.683777294771028280456455417607, −7.60456305043385297700609189345, −6.77363804204790951916141929682, −5.60176479208612598635811325615, −4.58135806730148975042476585864, −3.44398242675916364507231889887, −2.57754349323878532443990378699, −1.49259331755070657054960976739, 1.33559380984821791744595937164, 2.43286453421352369983903334032, 3.19051743338986231150515289551, 4.94141055707319698814194513999, 6.19646416430595402534916496518, 6.54539269060700441254923362587, 7.35936512831060898608653524703, 8.206058858776629560336806864044, 8.901305633137581719461786027445, 9.695591122787028894145652670326

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