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Properties

Label 1-1148-1148.163-r1-0-0
Degree $1$
Conductor $1148$
Sign $-0.0633 + 0.997i$
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  + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)11-s − 13-s + 15-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s + 27-s − 29-s + (0.5 − 0.866i)31-s + (−0.5 − 0.866i)33-s + (−0.5 − 0.866i)37-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)11-s − 13-s + 15-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s + 27-s − 29-s + (0.5 − 0.866i)31-s + (−0.5 − 0.866i)33-s + (−0.5 − 0.866i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.0633 + 0.997i$
Analytic conductor: \(123.369\)
Root analytic conductor: \(123.369\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1148,\ (1:\ ),\ -0.0633 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2957566945 + 0.3151204055i\)
\(L(\frac12)\) \(\approx\) \(0.2957566945 + 0.3151204055i\)
\(L(1)\) \(\approx\) \(0.6411015239 + 0.04068442112i\)
\(L(1)\) \(\approx\) \(0.6411015239 + 0.04068442112i\)

Euler product

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

−21.03207664686785185256195186320, −19.78135643807554120063718660918, −19.16879640979247305734585329332, −18.68987598548352095475635490316, −18.02372096135271137942459758004, −16.917874384284525600447638284554, −16.589630751208550348691659904447, −15.35153954465327467817012144958, −14.60992322494888029202186814913, −13.924828269043851873411266202262, −12.92295486673869570411202160572, −12.24763649023715308384372663687, −11.52709845556922364999088945917, −10.61769575695488941648260695221, −10.214114153431525976356794004784, −8.5947062501239019713802947923, −7.94982202336672884603302749799, −7.20912546567363025978527101257, −6.380834503002554474690112781622, −5.68217199951886491922703166218, −4.612319348499775218121723871931, −3.3442559255956144946057620261, −2.58542823533733647044930968032, −1.47409039776532746241019399023, −0.16115298690987952807583905184, 0.568977154952572374723702001654, 2.08827732599362342362549967604, 3.30518018000274349200188693888, 4.29059866006030020492369772207, 5.04511443979751149369128110421, 5.426617723629440067916312129108, 6.91078393364906877105845464348, 7.6220784593839827473185098300, 8.71752651358757708980659230054, 9.56880060327708124635930813481, 9.990137578338147606294738636849, 11.2360319963660772875526268746, 11.74459594516319000900914770760, 12.58839089129728695903422249631, 13.29917010688602519788003964042, 14.61447996983324141242982858779, 15.2772156194425507538973378877, 15.81878858496020994474584509503, 16.74629276070275528479727840640, 17.212195391842366011686971263076, 17.99632143939852950039777727623, 19.146780877004034477307640105738, 19.98013173727123237627034461426, 20.586046343272668295690505433240, 21.2378803550600614259586766825

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