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Properties

Label 1-1375-1375.1047-r0-0-0
Degree $1$
Conductor $1375$
Sign $0.709 + 0.704i$
Analytic cond. $6.38547$
Root an. cond. $6.38547$
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  + (0.904 + 0.425i)2-s + (−0.844 − 0.535i)3-s + (0.637 + 0.770i)4-s + (−0.535 − 0.844i)6-s + (0.587 + 0.809i)7-s + (0.248 + 0.968i)8-s + (0.425 + 0.904i)9-s + (−0.125 − 0.992i)12-s + (0.904 − 0.425i)13-s + (0.187 + 0.982i)14-s + (−0.187 + 0.982i)16-s + (−0.248 − 0.968i)17-s + i·18-s + (0.0627 − 0.998i)19-s + (−0.0627 − 0.998i)21-s + ⋯
L(s)  = 1  + (0.904 + 0.425i)2-s + (−0.844 − 0.535i)3-s + (0.637 + 0.770i)4-s + (−0.535 − 0.844i)6-s + (0.587 + 0.809i)7-s + (0.248 + 0.968i)8-s + (0.425 + 0.904i)9-s + (−0.125 − 0.992i)12-s + (0.904 − 0.425i)13-s + (0.187 + 0.982i)14-s + (−0.187 + 0.982i)16-s + (−0.248 − 0.968i)17-s + i·18-s + (0.0627 − 0.998i)19-s + (−0.0627 − 0.998i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.709 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.709 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.709 + 0.704i$
Analytic conductor: \(6.38547\)
Root analytic conductor: \(6.38547\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1047, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ 0.709 + 0.704i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.263923519 + 0.9335914440i\)
\(L(\frac12)\) \(\approx\) \(2.263923519 + 0.9335914440i\)
\(L(1)\) \(\approx\) \(1.563773709 + 0.3826599941i\)
\(L(1)\) \(\approx\) \(1.563773709 + 0.3826599941i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.904 + 0.425i)T \)
3 \( 1 + (-0.844 - 0.535i)T \)
7 \( 1 + (0.587 + 0.809i)T \)
13 \( 1 + (0.904 - 0.425i)T \)
17 \( 1 + (-0.248 - 0.968i)T \)
19 \( 1 + (0.0627 - 0.998i)T \)
23 \( 1 + (0.904 + 0.425i)T \)
29 \( 1 + (0.535 - 0.844i)T \)
31 \( 1 + (0.535 + 0.844i)T \)
37 \( 1 + (-0.481 + 0.876i)T \)
41 \( 1 + (0.992 - 0.125i)T \)
43 \( 1 + (-0.587 - 0.809i)T \)
47 \( 1 + (0.770 - 0.637i)T \)
53 \( 1 + (-0.844 - 0.535i)T \)
59 \( 1 + (0.425 + 0.904i)T \)
61 \( 1 + (0.187 + 0.982i)T \)
67 \( 1 + (-0.770 - 0.637i)T \)
71 \( 1 + (0.0627 + 0.998i)T \)
73 \( 1 + (0.982 - 0.187i)T \)
79 \( 1 + (-0.637 - 0.770i)T \)
83 \( 1 + (-0.770 - 0.637i)T \)
89 \( 1 + (0.992 + 0.125i)T \)
97 \( 1 + (-0.998 + 0.0627i)T \)
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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

−20.923622598361479831817945300880, −20.40221504350070471599014845397, −19.35405319740576527078863923820, −18.53780697836680173600955945420, −17.630268411736110194980642911161, −16.78785624256564628438742978302, −16.18158910684102470611446127079, −15.358088552540348218755705925179, −14.54944434932668804634298758806, −13.947858643908155155245655967164, −12.894534907135813276885322787690, −12.36067327542441049386310404583, −11.24694469772537985561428938090, −10.957136642003722113999377364970, −10.28962065371336054689136791058, −9.36818028404611206119697872195, −8.156634426286487409792723050997, −6.97495469486713586909520951182, −6.27367858269887847442756111297, −5.53816973751007492327857345307, −4.52853328303656156425764964748, −4.08526500519594179109313490982, −3.24345180438771589400985909406, −1.72271645513432950663583977574, −0.97468731750216534317166658735, 1.09813776483058210037265101516, 2.28111434063904069721524026488, 3.07874798073329502368669645578, 4.47490424245037440790395724523, 5.115880122936243607912995609203, 5.7407963795730964252783899006, 6.61554744191609854708269947144, 7.273708239454895092668992017562, 8.22023528408841882172459801525, 8.957532135315424489062218862040, 10.46822488396287145011464650950, 11.31751501569097397863190722417, 11.7317326653342347161965878603, 12.47666199701688747145813939705, 13.44743761712630602375110206567, 13.72984339995293180707654541540, 14.976586207193657720025962319602, 15.66744888174656079991442779909, 16.12839960364669582512214569895, 17.31281610119478717962795229650, 17.67304448715114093620083228472, 18.47790338848601600457531567468, 19.36461867685107246439070928698, 20.44065486415993493209109992079, 21.21256305625997585664015610811

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