Properties

Label 2-2548-7.4-c1-0-33
Degree $2$
Conductor $2548$
Sign $-0.386 + 0.922i$
Analytic cond. $20.3458$
Root an. cond. $4.51064$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 1.73i)3-s + (−0.5 − 0.866i)5-s + (−0.499 − 0.866i)9-s + (2 − 3.46i)11-s + 13-s − 1.99·15-s + (1 − 1.73i)17-s + (0.5 + 0.866i)19-s + (3.5 + 6.06i)23-s + (2 − 3.46i)25-s + 4.00·27-s − 5·29-s + (4.5 − 7.79i)31-s + (−3.99 − 6.92i)33-s + (1 + 1.73i)37-s + ⋯
L(s)  = 1  + (0.577 − 0.999i)3-s + (−0.223 − 0.387i)5-s + (−0.166 − 0.288i)9-s + (0.603 − 1.04i)11-s + 0.277·13-s − 0.516·15-s + (0.242 − 0.420i)17-s + (0.114 + 0.198i)19-s + (0.729 + 1.26i)23-s + (0.400 − 0.692i)25-s + 0.769·27-s − 0.928·29-s + (0.808 − 1.39i)31-s + (−0.696 − 1.20i)33-s + (0.164 + 0.284i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $-0.386 + 0.922i$
Analytic conductor: \(20.3458\)
Root analytic conductor: \(4.51064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2548} (1145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2548,\ (\ :1/2),\ -0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.240260509\)
\(L(\frac12)\) \(\approx\) \(2.240260509\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
13 \( 1 - T \)
good3 \( 1 + (-1 + 1.73i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.5 - 6.06i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 5T + 29T^{2} \)
31 \( 1 + (-4.5 + 7.79i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + (4.5 + 7.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7 + 12.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5 - 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 14T + 71T^{2} \)
73 \( 1 + (1.5 - 2.59i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.5 + 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 5T + 83T^{2} \)
89 \( 1 + (-4.5 - 7.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.576557333304831194821038880185, −7.86503275641902387232717258353, −7.33172431260927536375271535051, −6.41395353529576368830825707878, −5.71998758098083371334693730987, −4.67166847417759882508252958057, −3.62155780516508405477539417710, −2.82659703759217071234806557988, −1.65098321519603733686082269949, −0.75395473796557944257600146978, 1.40487108238202274894544768303, 2.78669386246033289589965643685, 3.46543002298796179756053140952, 4.38152068970085890237994398879, 4.85481365142780101930846655449, 6.12892221193525003179226695600, 6.91388535701494826473542892400, 7.59615207306682079744560021213, 8.680865787335957949403779501525, 9.092496202876055546141401646199

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