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Properties

Label 1-1380-1380.1343-r1-0-0
Degree $1$
Conductor $1380$
Sign $0.907 - 0.420i$
Analytic cond. $148.301$
Root an. cond. $148.301$
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)7-s + (0.841 + 0.540i)11-s + (−0.755 − 0.654i)13-s + (−0.909 + 0.415i)17-s + (0.415 − 0.909i)19-s + (0.415 + 0.909i)29-s + (0.142 + 0.989i)31-s + (−0.281 + 0.959i)37-s + (0.959 − 0.281i)41-s + (0.989 + 0.142i)43-s − i·47-s + (0.142 − 0.989i)49-s + (−0.755 + 0.654i)53-s + (0.654 − 0.755i)59-s + (−0.142 − 0.989i)61-s + ⋯
L(s)  = 1  + (0.755 − 0.654i)7-s + (0.841 + 0.540i)11-s + (−0.755 − 0.654i)13-s + (−0.909 + 0.415i)17-s + (0.415 − 0.909i)19-s + (0.415 + 0.909i)29-s + (0.142 + 0.989i)31-s + (−0.281 + 0.959i)37-s + (0.959 − 0.281i)41-s + (0.989 + 0.142i)43-s − i·47-s + (0.142 − 0.989i)49-s + (−0.755 + 0.654i)53-s + (0.654 − 0.755i)59-s + (−0.142 − 0.989i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.907 - 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.907 - 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.907 - 0.420i$
Analytic conductor: \(148.301\)
Root analytic conductor: \(148.301\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (1343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1380,\ (1:\ ),\ 0.907 - 0.420i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.403628340 - 0.5297367979i\)
\(L(\frac12)\) \(\approx\) \(2.403628340 - 0.5297367979i\)
\(L(1)\) \(\approx\) \(1.220511005 - 0.09906562925i\)
\(L(1)\) \(\approx\) \(1.220511005 - 0.09906562925i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good7 \( 1 + (0.755 - 0.654i)T \)
11 \( 1 + (0.841 + 0.540i)T \)
13 \( 1 + (-0.755 - 0.654i)T \)
17 \( 1 + (-0.909 + 0.415i)T \)
19 \( 1 + (0.415 - 0.909i)T \)
29 \( 1 + (0.415 + 0.909i)T \)
31 \( 1 + (0.142 + 0.989i)T \)
37 \( 1 + (-0.281 + 0.959i)T \)
41 \( 1 + (0.959 - 0.281i)T \)
43 \( 1 + (0.989 + 0.142i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.755 + 0.654i)T \)
59 \( 1 + (0.654 - 0.755i)T \)
61 \( 1 + (-0.142 - 0.989i)T \)
67 \( 1 + (0.540 + 0.841i)T \)
71 \( 1 + (0.841 - 0.540i)T \)
73 \( 1 + (0.909 + 0.415i)T \)
79 \( 1 + (-0.654 + 0.755i)T \)
83 \( 1 + (0.281 - 0.959i)T \)
89 \( 1 + (-0.142 + 0.989i)T \)
97 \( 1 + (0.281 + 0.959i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.95144048300916735156449146264, −19.80720002442374033700229821170, −19.21813036154449871322399186978, −18.45344212195929430660071489742, −17.63544615606666315746900163905, −17.01112611716347683676296296452, −16.11658003248754656141224309521, −15.39114285636151252751467771130, −14.28812701887065721169528177350, −14.232021983031619646575223872843, −12.994077198957381360574864556442, −12.013456661045778517283345355407, −11.58033941477103989480785005900, −10.81600854401868073072632859177, −9.56442918219721926397779789795, −9.12233805152651661049123592474, −8.16245300155857343315900452604, −7.4090042606839790133842629069, −6.34935782036962772241108971216, −5.66280137496296048599342099152, −4.60883322057676606150718968969, −3.95467489310904157698781649964, −2.6249780630831029009973091465, −1.916055161408180304857316777318, −0.75171611192356304749262877119, 0.65353421741015133197059210526, 1.597229932223154039123712635448, 2.62788691230599128827137569345, 3.756210904544291483093691708052, 4.65780679343009044768602469133, 5.21063122103093692245551991367, 6.602503207306607097162798340557, 7.1092258191014168795793112789, 8.02576218017108718429622424021, 8.87453257293145823660242868113, 9.72385829581033691141293922212, 10.624677523493685412054428369311, 11.2328170144069697893209244644, 12.16877223292767965284723562828, 12.86103451018582650044696678658, 13.86783651324446695469526982917, 14.44317391704678700047152549146, 15.21712332891467786434647696205, 15.96080276498552159532602329173, 17.14337325102051770886230838519, 17.463159215193355539804742237586, 18.07037383914647838649705274037, 19.28330422707901805580505168173, 20.06486106633895687296409882436, 20.24783929502133546225936045646

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