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Properties

Label 2-2548-91.4-c1-0-15
Degree $2$
Conductor $2548$
Sign $-0.478 - 0.878i$
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  + (0.937 + 1.62i)3-s + (−0.192 + 0.111i)5-s + (−0.258 + 0.447i)9-s + (−2.47 + 1.42i)11-s + (2.28 + 2.79i)13-s + (−0.361 − 0.208i)15-s + 2.43·17-s + (−0.488 − 0.281i)19-s + 4.09·23-s + (−2.47 + 4.28i)25-s + 4.65·27-s + (−1.79 + 3.10i)29-s + (−1.67 − 0.965i)31-s + (−4.64 − 2.67i)33-s − 1.79i·37-s + ⋯
L(s)  = 1  + (0.541 + 0.937i)3-s + (−0.0862 + 0.0497i)5-s + (−0.0861 + 0.149i)9-s + (−0.746 + 0.430i)11-s + (0.632 + 0.774i)13-s + (−0.0933 − 0.0539i)15-s + 0.590·17-s + (−0.112 − 0.0646i)19-s + 0.854·23-s + (−0.495 + 0.857i)25-s + 0.896·27-s + (−0.332 + 0.575i)29-s + (−0.300 − 0.173i)31-s + (−0.808 − 0.466i)33-s − 0.295i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.478 - 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.478 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $-0.478 - 0.878i$
Analytic conductor: \(20.3458\)
Root analytic conductor: \(4.51064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2548} (1733, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2548,\ (\ :1/2),\ -0.478 - 0.878i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.916682452\)
\(L(\frac12)\) \(\approx\) \(1.916682452\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.28 - 2.79i)T \)
good3 \( 1 + (-0.937 - 1.62i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.192 - 0.111i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.47 - 1.42i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 2.43T + 17T^{2} \)
19 \( 1 + (0.488 + 0.281i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.09T + 23T^{2} \)
29 \( 1 + (1.79 - 3.10i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.67 + 0.965i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 1.79iT - 37T^{2} \)
41 \( 1 + (-0.880 - 0.508i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.80 - 4.86i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.53 - 2.62i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.15 - 2.00i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 3.29iT - 59T^{2} \)
61 \( 1 + (-1.00 + 1.74i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (11.4 - 6.63i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (9.47 - 5.46i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.10 - 2.36i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.58 - 9.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.02iT - 83T^{2} \)
89 \( 1 + 10.1iT - 89T^{2} \)
97 \( 1 + (1.08 - 0.623i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.227147498229802280192216816761, −8.587520792793551549323589637371, −7.66283024498774651411861985566, −6.99415237785272383917100824053, −5.95913640174513649574937639193, −5.06906333374873522789632108237, −4.29576954706549514327517323466, −3.53439154203373967919850135260, −2.73799387703131930944259040384, −1.42179803630496564881279873414, 0.60626268667518479797468715670, 1.78489711416785641614913880947, 2.78451977934121546087804123291, 3.52318945762117226482259160955, 4.75978034425091494431359661391, 5.65077878157250788537838569264, 6.38737972906629133986256617250, 7.35015955663161497284830869884, 7.912173620299825427660258428727, 8.391547617629523439078838490315

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