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Properties

Label 1-1148-1148.251-r0-0-0
Degree $1$
Conductor $1148$
Sign $0.856 - 0.515i$
Analytic cond. $5.33128$
Root an. cond. $5.33128$
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  − i·3-s + (−0.809 − 0.587i)5-s − 9-s + (0.587 + 0.809i)11-s + (−0.951 − 0.309i)13-s + (0.587 − 0.809i)15-s + (−0.587 − 0.809i)17-s + (0.951 − 0.309i)19-s + (−0.309 + 0.951i)23-s + (0.309 + 0.951i)25-s − i·27-s + (0.587 − 0.809i)29-s + (−0.809 + 0.587i)31-s + (−0.809 + 0.587i)33-s + (−0.809 − 0.587i)37-s + ⋯
L(s)  = 1  − i·3-s + (−0.809 − 0.587i)5-s − 9-s + (0.587 + 0.809i)11-s + (−0.951 − 0.309i)13-s + (0.587 − 0.809i)15-s + (−0.587 − 0.809i)17-s + (0.951 − 0.309i)19-s + (−0.309 + 0.951i)23-s + (0.309 + 0.951i)25-s − i·27-s + (0.587 − 0.809i)29-s + (−0.809 + 0.587i)31-s + (−0.809 + 0.587i)33-s + (−0.809 − 0.587i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.856 - 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.856 - 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.856 - 0.515i$
Analytic conductor: \(5.33128\)
Root analytic conductor: \(5.33128\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1148,\ (0:\ ),\ 0.856 - 0.515i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8398671117 - 0.2332812054i\)
\(L(\frac12)\) \(\approx\) \(0.8398671117 - 0.2332812054i\)
\(L(1)\) \(\approx\) \(0.8164902782 + 0.1058553971i\)
\(L(1)\) \(\approx\) \(0.8164902782 + 0.1058553971i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good3 \( 1 - iT \)
5 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (0.587 + 0.809i)T \)
13 \( 1 + (-0.951 - 0.309i)T \)
17 \( 1 + (-0.587 - 0.809i)T \)
19 \( 1 + (0.951 - 0.309i)T \)
23 \( 1 + (-0.309 + 0.951i)T \)
29 \( 1 + (0.587 - 0.809i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (0.309 - 0.951i)T \)
47 \( 1 + (-0.951 - 0.309i)T \)
53 \( 1 + (0.587 - 0.809i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + (0.587 - 0.809i)T \)
71 \( 1 + (0.587 + 0.809i)T \)
73 \( 1 + T \)
79 \( 1 - iT \)
83 \( 1 + T \)
89 \( 1 + (0.951 - 0.309i)T \)
97 \( 1 + (0.587 - 0.809i)T \)
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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

−21.55475096371479599409844659559, −20.143794518041306947336793081538, −19.77109028740721770116412306167, −19.02063900280071970352657996466, −18.46165570257920580434548378267, −17.643562670947781781642192711148, −16.74960352709105267180591068194, −16.07076403517434683496159236789, −14.8448993331016477361833262078, −14.413249192407159041881750118531, −13.632084779314625830401294498071, −12.55881582701863030967623838701, −12.001225040737193105296005489580, −11.27913022819033466584884003851, −10.53519488517919310334084567821, −9.27186753519130386888593849487, −8.348074109382009728435188233109, −7.739845464098574625838809497331, −6.79489095020634569956790177389, −6.342208467937112140662443849390, −5.16084228961087216877604883910, −3.95582170596957045841447060864, −3.090396145270944404342331041334, −2.173259027672196772106131334159, −0.95630822847314633570000870017, 0.43588183165754962896227354847, 2.064316182445676487217594634264, 3.27767841962066277464893560088, 4.026731251073224270884541343206, 4.93734104332770097686933234270, 5.345703234960909947943991605146, 6.89175431835250647873798667021, 7.60538828438410719300864798243, 8.620957199661073397056657467987, 9.42904611044897140396365301875, 9.904674772234647401581143964312, 11.075763843606384342951699782752, 11.80721568659902857714842028697, 12.30546116768205939782871897928, 13.51740320500642655825496812097, 14.41108527837684561396303362530, 15.21289540376620614523000336769, 15.78306891019411828747232171431, 16.398714093183878738878021395495, 17.37507100602538182219745628394, 17.81927533730471111989453536303, 19.287349562989986643057993139998, 19.94249510467565679553360034623, 20.27014238786902433159926833374, 21.179180382115659399598303313305

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