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Properties

Label 2-2548-91.16-c1-0-43
Degree $2$
Conductor $2548$
Sign $-0.927 + 0.374i$
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 + (1.5 − 2.59i)5-s + (−0.499 − 0.866i)9-s + (−3.5 + 0.866i)13-s + (−3 − 5.19i)15-s + 3·17-s + (−1 − 1.73i)19-s − 6·23-s + (−2 − 3.46i)25-s + 4.00·27-s + (−4.5 − 7.79i)29-s + (−1 − 1.73i)31-s − 7·37-s + (−2 + 6.92i)39-s + (−1.5 − 2.59i)41-s + ⋯
L(s)  = 1  + (0.577 − 0.999i)3-s + (0.670 − 1.16i)5-s + (−0.166 − 0.288i)9-s + (−0.970 + 0.240i)13-s + (−0.774 − 1.34i)15-s + 0.727·17-s + (−0.229 − 0.397i)19-s − 1.25·23-s + (−0.400 − 0.692i)25-s + 0.769·27-s + (−0.835 − 1.44i)29-s + (−0.179 − 0.311i)31-s − 1.15·37-s + (−0.320 + 1.10i)39-s + (−0.234 − 0.405i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 + 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $-0.927 + 0.374i$
Analytic conductor: \(20.3458\)
Root analytic conductor: \(4.51064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2548} (1745, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2548,\ (\ :1/2),\ -0.927 + 0.374i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.959433275\)
\(L(\frac12)\) \(\approx\) \(1.959433275\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.5 - 0.866i)T \)
good3 \( 1 + (-1 + 1.73i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 + (4.5 + 7.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 7T + 37T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.5 + 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + (2.5 + 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3 + 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (7 - 12.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.452664844453679239189769093960, −7.85696181145068780415712598843, −7.21118423529428943881744979069, −6.28158311034488783967551418528, −5.43536537940392242209814991570, −4.74056786212317888687980613169, −3.64901296651492747226524531439, −2.21151771073269309167050483215, −1.88210027625719964194518484341, −0.54985498454079835692019597952, 1.79229660327676767780509767249, 2.87204148640304126630083330264, 3.39934445629807579001729960452, 4.35624408220353777111013392431, 5.32636932908596543647770963037, 6.10141516748893492264726136130, 6.99470481075728621971243068380, 7.68043368172599761131179078276, 8.619464157148427778335960365904, 9.497489877109192395049182101284

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