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Properties

Label 2-2548-13.3-c1-0-20
Degree $2$
Conductor $2548$
Sign $0.522 + 0.852i$
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  − 3·5-s + (1.5 − 2.59i)9-s + (1 + 1.73i)11-s + (2.5 − 2.59i)13-s + (−3.5 + 6.06i)17-s + (1 − 1.73i)19-s + (2 + 3.46i)23-s + 4·25-s + (−0.5 − 0.866i)29-s − 4·31-s + (−0.5 − 0.866i)37-s + (−1.5 − 2.59i)41-s + (3 − 5.19i)43-s + (−4.5 + 7.79i)45-s + 10·47-s + ⋯
L(s)  = 1  − 1.34·5-s + (0.5 − 0.866i)9-s + (0.301 + 0.522i)11-s + (0.693 − 0.720i)13-s + (−0.848 + 1.47i)17-s + (0.229 − 0.397i)19-s + (0.417 + 0.722i)23-s + 0.800·25-s + (−0.0928 − 0.160i)29-s − 0.718·31-s + (−0.0821 − 0.142i)37-s + (−0.234 − 0.405i)41-s + (0.457 − 0.792i)43-s + (−0.670 + 1.16i)45-s + 1.45·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $0.522 + 0.852i$
Analytic conductor: \(20.3458\)
Root analytic conductor: \(4.51064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2548} (393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2548,\ (\ :1/2),\ 0.522 + 0.852i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.212465437\)
\(L(\frac12)\) \(\approx\) \(1.212465437\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 + 2.59i)T \)
good3 \( 1 + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 3T + 5T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.5 - 6.06i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3 + 5.19i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 10T + 47T^{2} \)
53 \( 1 + 7T + 53T^{2} \)
59 \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4 + 6.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3 + 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 11T + 73T^{2} \)
79 \( 1 + 14T + 79T^{2} \)
83 \( 1 - 14T + 83T^{2} \)
89 \( 1 + (5 + 8.66i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1 + 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.771533196099515226546872989751, −7.963161202791244164806841281602, −7.29976089567537064456536202350, −6.60547740762340882038418690499, −5.73652921882286817945591531776, −4.58671649043391313875551568453, −3.80506928903124666121220395684, −3.42700793818715616641761581894, −1.82193537847506765450927324581, −0.52528546197402985772811021165, 0.960025583495519507257773129472, 2.37465898505501122014658200883, 3.46287247027335276241163590520, 4.28696826401480677940843569730, 4.83395384720296235533461168517, 5.96888221250178225841222573394, 7.04847782351937394323935321024, 7.35503276788284135711758507685, 8.318224048353224040721873255330, 8.834407231177779058629724143036

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