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Properties

Label 1-2668-2668.215-r0-0-0
Degree $1$
Conductor $2668$
Sign $0.200 + 0.979i$
Analytic cond. $12.3901$
Root an. cond. $12.3901$
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)3-s + (0.142 − 0.989i)5-s + (−0.415 − 0.909i)7-s + (0.142 + 0.989i)9-s + (0.281 + 0.959i)11-s + (−0.415 + 0.909i)13-s + (0.755 − 0.654i)15-s + (0.540 + 0.841i)17-s + (0.540 − 0.841i)19-s + (0.281 − 0.959i)21-s + (−0.959 − 0.281i)25-s + (−0.540 + 0.841i)27-s + (−0.755 + 0.654i)31-s + (−0.415 + 0.909i)33-s + (−0.959 + 0.281i)35-s + ⋯
L(s)  = 1  + (0.755 + 0.654i)3-s + (0.142 − 0.989i)5-s + (−0.415 − 0.909i)7-s + (0.142 + 0.989i)9-s + (0.281 + 0.959i)11-s + (−0.415 + 0.909i)13-s + (0.755 − 0.654i)15-s + (0.540 + 0.841i)17-s + (0.540 − 0.841i)19-s + (0.281 − 0.959i)21-s + (−0.959 − 0.281i)25-s + (−0.540 + 0.841i)27-s + (−0.755 + 0.654i)31-s + (−0.415 + 0.909i)33-s + (−0.959 + 0.281i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2668 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.200 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2668 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.200 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2668\)    =    \(2^{2} \cdot 23 \cdot 29\)
Sign: $0.200 + 0.979i$
Analytic conductor: \(12.3901\)
Root analytic conductor: \(12.3901\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2668} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2668,\ (0:\ ),\ 0.200 + 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.446414901 + 1.180978937i\)
\(L(\frac12)\) \(\approx\) \(1.446414901 + 1.180978937i\)
\(L(1)\) \(\approx\) \(1.267690354 + 0.2603565987i\)
\(L(1)\) \(\approx\) \(1.267690354 + 0.2603565987i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good3 \( 1 + (0.755 + 0.654i)T \)
5 \( 1 + (0.142 - 0.989i)T \)
7 \( 1 + (-0.415 - 0.909i)T \)
11 \( 1 + (0.281 + 0.959i)T \)
13 \( 1 + (-0.415 + 0.909i)T \)
17 \( 1 + (0.540 + 0.841i)T \)
19 \( 1 + (0.540 - 0.841i)T \)
31 \( 1 + (-0.755 + 0.654i)T \)
37 \( 1 + (-0.989 + 0.142i)T \)
41 \( 1 + (0.989 + 0.142i)T \)
43 \( 1 + (0.755 + 0.654i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.415 + 0.909i)T \)
59 \( 1 + (-0.415 + 0.909i)T \)
61 \( 1 + (0.755 - 0.654i)T \)
67 \( 1 + (-0.959 - 0.281i)T \)
71 \( 1 + (-0.959 - 0.281i)T \)
73 \( 1 + (0.540 - 0.841i)T \)
79 \( 1 + (-0.909 - 0.415i)T \)
83 \( 1 + (0.142 + 0.989i)T \)
89 \( 1 + (-0.755 - 0.654i)T \)
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

−19.09542547364480492294063917655, −18.50303533578618496899908623367, −18.11232479248785015261642910438, −17.22616782038432379586703208219, −16.1234421920861221084044728370, −15.54656579325570354125974303282, −14.640388730358277173022547240889, −14.34398889435041175184845001116, −13.53836204904843869860488754406, −12.807797172101193771644088412084, −12.00626822735910398672721131658, −11.48108507902373055731635991437, −10.37217553156191852706298355712, −9.70163023898524009874968013961, −8.987246649702229008163974314186, −8.18762729437600018653772366497, −7.44234771205679457491870915648, −6.84223732916226163936881563870, −5.75900126368144270559661874783, −5.59661592942324606831064612499, −3.79161184253032401681108120134, −3.198892697418771209004233180370, −2.65390063905666856081980063728, −1.830542800709558348159208610614, −0.52870609370548858608582162133, 1.19998892752844577507163511783, 1.93068186057762126176583054920, 3.02501839902886962798295796179, 3.99515919294868713023961452978, 4.453134418839707751736115651856, 5.11614993374168952186455212024, 6.24209246574760861399055181676, 7.35054305990413377594020425223, 7.686749658708201033666823224503, 8.974302201224985203595082430081, 9.181430406791913796036569955468, 9.98893296499598620239894941024, 10.571181809300766021002834707685, 11.60602103117313427592237054499, 12.57047801255809638844669278548, 13.015558899672652701408257859920, 13.98952526772967135310610239337, 14.314495881869917099802456292155, 15.28872283129937077803426252735, 15.99796536220538593638067646138, 16.60825229156851369464668253925, 17.14841697082303484492534336417, 17.85911896451033969542023490550, 19.18106100486449284985463655269, 19.6619797373762995743109862649

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