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Properties

Label 1-1148-1148.1147-r0-0-0
Degree $1$
Conductor $1148$
Sign $1$
Analytic cond. $5.33128$
Root an. cond. $5.33128$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s + 9-s + 11-s + 13-s + 15-s + 17-s − 19-s − 23-s + 25-s − 27-s − 29-s + 31-s − 33-s + 37-s − 39-s − 43-s − 45-s − 47-s − 51-s − 53-s − 55-s + 57-s + 59-s − 61-s − 65-s + 67-s + ⋯
L(s)  = 1  − 3-s − 5-s + 9-s + 11-s + 13-s + 15-s + 17-s − 19-s − 23-s + 25-s − 27-s − 29-s + 31-s − 33-s + 37-s − 39-s − 43-s − 45-s − 47-s − 51-s − 53-s − 55-s + 57-s + 59-s − 61-s − 65-s + 67-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $1$
Analytic conductor: \(5.33128\)
Root analytic conductor: \(5.33128\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1148} (1147, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 1148,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9132237921\)
\(L(\frac12)\) \(\approx\) \(0.9132237921\)
\(L(1)\) \(\approx\) \(0.7503779886\)
\(L(1)\) \(\approx\) \(0.7503779886\)

Euler product

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

−21.395400260215964572418489795782, −20.56540371450547115137984980887, −19.64249966480912472365623538878, −18.883344595689330269823569136256, −18.32118761330236334033417234263, −17.30148301429413037946829569061, −16.59698329293935501106756904521, −16.07125783923858819174720045805, −15.19142857651630923850528421799, −14.45947605168858943290584306041, −13.2836902220496356970819318459, −12.46548305823402100720057357517, −11.73948405025658552623162810310, −11.26551106254483217459168765751, −10.397540225264075492069959629797, −9.484880950440736484547650288944, −8.35790174887817223658843703418, −7.67289383837640347987866915967, −6.54685571688354696853427334719, −6.11605171693459882108355910101, −4.91762164290475360566942484658, −4.04998764575539115736630338004, −3.472940234776751711590552281902, −1.74461496277952618414248122792, −0.73433715700770444330418551446, 0.73433715700770444330418551446, 1.74461496277952618414248122792, 3.472940234776751711590552281902, 4.04998764575539115736630338004, 4.91762164290475360566942484658, 6.11605171693459882108355910101, 6.54685571688354696853427334719, 7.67289383837640347987866915967, 8.35790174887817223658843703418, 9.484880950440736484547650288944, 10.397540225264075492069959629797, 11.26551106254483217459168765751, 11.73948405025658552623162810310, 12.46548305823402100720057357517, 13.2836902220496356970819318459, 14.45947605168858943290584306041, 15.19142857651630923850528421799, 16.07125783923858819174720045805, 16.59698329293935501106756904521, 17.30148301429413037946829569061, 18.32118761330236334033417234263, 18.883344595689330269823569136256, 19.64249966480912472365623538878, 20.56540371450547115137984980887, 21.395400260215964572418489795782

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