Properties

Label 2-2548-7.2-c1-0-16
Degree $2$
Conductor $2548$
Sign $0.968 - 0.250i$
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)5-s + (1.5 − 2.59i)9-s + (1 + 1.73i)11-s + 13-s + (3 + 5.19i)17-s + (−3 + 5.19i)19-s + (−4 + 6.92i)23-s + (0.500 + 0.866i)25-s + 2·29-s + (5 + 8.66i)31-s + (3 − 5.19i)37-s + 6·41-s + 4·43-s + (−3 − 5.19i)45-s + (−1 + 1.73i)47-s + ⋯
L(s)  = 1  + (0.447 − 0.774i)5-s + (0.5 − 0.866i)9-s + (0.301 + 0.522i)11-s + 0.277·13-s + (0.727 + 1.26i)17-s + (−0.688 + 1.19i)19-s + (−0.834 + 1.44i)23-s + (0.100 + 0.173i)25-s + 0.371·29-s + (0.898 + 1.55i)31-s + (0.493 − 0.854i)37-s + 0.937·41-s + 0.609·43-s + (−0.447 − 0.774i)45-s + (−0.145 + 0.252i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $0.968 - 0.250i$
Analytic conductor: \(20.3458\)
Root analytic conductor: \(4.51064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2548} (1353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2548,\ (\ :1/2),\ 0.968 - 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.118350900\)
\(L(\frac12)\) \(\approx\) \(2.118350900\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3 - 5.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4 - 6.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + (-5 - 8.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3 + 5.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (1 - 1.73i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5 + 8.66i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5 + 8.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10T + 71T^{2} \)
73 \( 1 + (-1 - 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−9.010420766079092288634322711011, −8.193539058565413101320101811553, −7.52868990801572514846503236074, −6.37780283572819713656672817310, −5.98341170242348449409120938542, −5.01357180371621334163280220555, −4.03283452455296426575231494459, −3.46814096634820451459141665515, −1.81155707157280724890047392864, −1.20918338678066104249379417414, 0.806988999959568857209816037824, 2.40852956822482172293138287791, 2.77669978438021523427971363561, 4.21496948097574950025193096025, 4.80530351955309641219575176789, 5.99433791467696264493881126675, 6.46802647414106140048907437236, 7.34004442374994101776155842679, 8.041417319992677436093810672547, 8.863600638933600365605831257342

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