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Properties

Label 1-1380-1380.383-r0-0-0
Degree $1$
Conductor $1380$
Sign $0.758 - 0.651i$
Analytic cond. $6.40869$
Root an. cond. $6.40869$
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)7-s + (0.959 − 0.281i)11-s + (0.909 + 0.415i)13-s + (−0.540 − 0.841i)17-s + (−0.841 − 0.540i)19-s + (0.841 − 0.540i)29-s + (0.654 + 0.755i)31-s + (0.989 − 0.142i)37-s + (0.142 − 0.989i)41-s + (−0.755 − 0.654i)43-s − i·47-s + (0.654 − 0.755i)49-s + (−0.909 + 0.415i)53-s + (−0.415 + 0.909i)59-s + (0.654 + 0.755i)61-s + ⋯
L(s)  = 1  + (0.909 − 0.415i)7-s + (0.959 − 0.281i)11-s + (0.909 + 0.415i)13-s + (−0.540 − 0.841i)17-s + (−0.841 − 0.540i)19-s + (0.841 − 0.540i)29-s + (0.654 + 0.755i)31-s + (0.989 − 0.142i)37-s + (0.142 − 0.989i)41-s + (−0.755 − 0.654i)43-s − i·47-s + (0.654 − 0.755i)49-s + (−0.909 + 0.415i)53-s + (−0.415 + 0.909i)59-s + (0.654 + 0.755i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.758 - 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.758 - 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.758 - 0.651i$
Analytic conductor: \(6.40869\)
Root analytic conductor: \(6.40869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1380,\ (0:\ ),\ 0.758 - 0.651i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.782744654 - 0.6609533343i\)
\(L(\frac12)\) \(\approx\) \(1.782744654 - 0.6609533343i\)
\(L(1)\) \(\approx\) \(1.271321395 - 0.1766680345i\)
\(L(1)\) \(\approx\) \(1.271321395 - 0.1766680345i\)

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 \( 1 + (0.909 - 0.415i)T \)
11 \( 1 + (0.959 - 0.281i)T \)
13 \( 1 + (0.909 + 0.415i)T \)
17 \( 1 + (-0.540 - 0.841i)T \)
19 \( 1 + (-0.841 - 0.540i)T \)
29 \( 1 + (0.841 - 0.540i)T \)
31 \( 1 + (0.654 + 0.755i)T \)
37 \( 1 + (0.989 - 0.142i)T \)
41 \( 1 + (0.142 - 0.989i)T \)
43 \( 1 + (-0.755 - 0.654i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.909 + 0.415i)T \)
59 \( 1 + (-0.415 + 0.909i)T \)
61 \( 1 + (0.654 + 0.755i)T \)
67 \( 1 + (-0.281 + 0.959i)T \)
71 \( 1 + (-0.959 - 0.281i)T \)
73 \( 1 + (-0.540 + 0.841i)T \)
79 \( 1 + (-0.415 + 0.909i)T \)
83 \( 1 + (-0.989 + 0.142i)T \)
89 \( 1 + (0.654 - 0.755i)T \)
97 \( 1 + (-0.989 - 0.142i)T \)
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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

−20.92446557389269509884429307239, −20.21773068291578037168102355726, −19.415980757998814090992528474993, −18.62332358531208574357302607898, −17.79610352943040922472352673449, −17.30555103519471452042322226939, −16.41552786794487990374106762616, −15.46017372496122599818802999925, −14.79795327737844045526886159159, −14.246578886323289312978414141959, −13.17712555807792793621367686918, −12.507147808160073325090408267020, −11.54743779529481773552052358865, −11.028874547739107267297931240918, −10.11246488541438885849472131848, −9.1133185214769006140113923842, −8.344200268001734440069889913775, −7.83766583851538428929827885615, −6.36914402942307438319949027121, −6.146143924238878046691712691484, −4.77848580470291054338300143057, −4.20493397859257291374701072419, −3.12765643377692255160948829167, −1.92106521232701527406665125175, −1.229952690565801741581927803820, 0.841940969851622532344541859699, 1.76132920819992205716793181637, 2.85653952025051541466290112951, 4.113736230811502103904190165851, 4.503011239538670790776554459346, 5.68235373078740260058117699003, 6.6346710108747322979110676552, 7.23247955747866526327492308624, 8.560910620667592059335433257306, 8.7137400849840235825609947746, 9.92359000245646046693946713760, 10.85844386226168640150035332519, 11.458212761146637416815574854, 12.07923745776228840135651627903, 13.34997620562481431072463050055, 13.84345843050983794365307641943, 14.55460464703732572251391453231, 15.43002003981198088455017010956, 16.22252511847903172013006893780, 17.07786227979332142207385283165, 17.6578704861020423543803864726, 18.425316316095917388562945837350, 19.30043631625127713960856014350, 20.01531969309293617926467787765, 20.77408431809465432905618170989

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