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Properties

Label 1-1148-1148.383-r1-0-0
Degree $1$
Conductor $1148$
Sign $-0.208 + 0.977i$
Analytic cond. $123.369$
Root an. cond. $123.369$
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  + (−0.258 − 0.965i)3-s + (0.866 − 0.5i)5-s + (−0.866 + 0.5i)9-s + (−0.258 − 0.965i)11-s + (0.707 − 0.707i)13-s + (−0.707 − 0.707i)15-s + (−0.965 + 0.258i)17-s + (−0.258 + 0.965i)19-s + (−0.5 − 0.866i)23-s + (0.5 − 0.866i)25-s + (0.707 + 0.707i)27-s + (−0.707 + 0.707i)29-s + (0.5 − 0.866i)31-s + (−0.866 + 0.5i)33-s + (−0.5 − 0.866i)37-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)3-s + (0.866 − 0.5i)5-s + (−0.866 + 0.5i)9-s + (−0.258 − 0.965i)11-s + (0.707 − 0.707i)13-s + (−0.707 − 0.707i)15-s + (−0.965 + 0.258i)17-s + (−0.258 + 0.965i)19-s + (−0.5 − 0.866i)23-s + (0.5 − 0.866i)25-s + (0.707 + 0.707i)27-s + (−0.707 + 0.707i)29-s + (0.5 − 0.866i)31-s + (−0.866 + 0.5i)33-s + (−0.5 − 0.866i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.208 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.208 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.208 + 0.977i$
Analytic conductor: \(123.369\)
Root analytic conductor: \(123.369\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1148,\ (1:\ ),\ -0.208 + 0.977i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1949852269 - 0.2409670662i\)
\(L(\frac12)\) \(\approx\) \(-0.1949852269 - 0.2409670662i\)
\(L(1)\) \(\approx\) \(0.7995491134 - 0.4485317653i\)
\(L(1)\) \(\approx\) \(0.7995491134 - 0.4485317653i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good3 \( 1 + (-0.258 - 0.965i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (-0.258 - 0.965i)T \)
13 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 + (-0.965 + 0.258i)T \)
19 \( 1 + (-0.258 + 0.965i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.707 + 0.707i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.258 + 0.965i)T \)
53 \( 1 + (0.258 + 0.965i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.866 + 0.5i)T \)
67 \( 1 + (-0.965 + 0.258i)T \)
71 \( 1 + (0.707 - 0.707i)T \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (0.965 + 0.258i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.965 - 0.258i)T \)
97 \( 1 + (-0.707 - 0.707i)T \)
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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

−21.5841407470965397815643331873, −21.08737611897520626063624106633, −20.296515625128131203911157757356, −19.48267842666048616789207752527, −18.30108418592944184604939301054, −17.702912490098441399593681387087, −17.16311901079123596760771422831, −16.15752527647111928685444590233, −15.39037312235736754250463855079, −14.88096343678024771316210328651, −13.78188122153769663714606215733, −13.353984298692191866963020847380, −12.05284112228494868889905809227, −11.191860047587312977200394379675, −10.6391086962514282893435453007, −9.61423917034868913157439518516, −9.34521285355852944034045018087, −8.24705012514957962829491348821, −6.85119293767069224285797997750, −6.38366217438978576374873934521, −5.26881697707470329688992698440, −4.60840025547175871064196233298, −3.61585159265379872654377602635, −2.56542592839996538719483027579, −1.66327971172344084359555066749, 0.06437612099801670272621066709, 1.04952287852179171215408457386, 1.95321200286622608576159600345, 2.84363648697580577323847675895, 4.11896893272402885705682465594, 5.439329006172334532071344083517, 5.925929952847185374140172368160, 6.57629699447232170086770727304, 7.8325988468308430303312268396, 8.47321369379578219419832447078, 9.17046863901125960288720360116, 10.5751874666760299898795256991, 10.883461787092475388389961467513, 12.17088789683707729192201627431, 12.7348024117123767756124430464, 13.557197066056057728110970997954, 13.89973937600317256140623779048, 15.019809893230810130804907961933, 16.20491621672764345963814235110, 16.75087443776685594641075262626, 17.56987506596037096252483583758, 18.26711440878339140134243999761, 18.77650598495874122704896893531, 19.80297810608978235031336585169, 20.5376642983430870832574856396

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