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Properties

Label 2-2548-91.81-c1-0-16
Degree $2$
Conductor $2548$
Sign $0.741 + 0.671i$
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  − 2.52·3-s + (−1.33 − 2.31i)5-s + 3.38·9-s − 2.14·11-s + (−2.05 + 2.96i)13-s + (3.37 + 5.83i)15-s + (−1.96 − 3.40i)17-s + 5.07·19-s + (−3.48 + 6.03i)23-s + (−1.06 + 1.83i)25-s − 0.960·27-s + (0.0611 + 0.106i)29-s + (−1.96 + 3.40i)31-s + 5.41·33-s + (−0.724 + 1.25i)37-s + ⋯
L(s)  = 1  − 1.45·3-s + (−0.596 − 1.03i)5-s + 1.12·9-s − 0.646·11-s + (−0.570 + 0.821i)13-s + (0.870 + 1.50i)15-s + (−0.476 − 0.825i)17-s + 1.16·19-s + (−0.726 + 1.25i)23-s + (−0.212 + 0.367i)25-s − 0.184·27-s + (0.0113 + 0.0196i)29-s + (−0.353 + 0.612i)31-s + 0.942·33-s + (−0.119 + 0.206i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.741 + 0.671i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.741 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $0.741 + 0.671i$
Analytic conductor: \(20.3458\)
Root analytic conductor: \(4.51064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2548} (1537, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2548,\ (\ :1/2),\ 0.741 + 0.671i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5498748078\)
\(L(\frac12)\) \(\approx\) \(0.5498748078\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.05 - 2.96i)T \)
good3 \( 1 + 2.52T + 3T^{2} \)
5 \( 1 + (1.33 + 2.31i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 2.14T + 11T^{2} \)
17 \( 1 + (1.96 + 3.40i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 5.07T + 19T^{2} \)
23 \( 1 + (3.48 - 6.03i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.0611 - 0.106i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.96 - 3.40i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.724 - 1.25i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.29 - 3.97i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.997 - 1.72i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.35 - 7.54i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.0515 + 0.0892i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.60 + 7.97i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 13.0T + 61T^{2} \)
67 \( 1 + 7.27T + 67T^{2} \)
71 \( 1 + (-5.76 + 9.98i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.53 - 7.86i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.409 - 0.708i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 13.1T + 83T^{2} \)
89 \( 1 + (-7.50 + 13.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.49 + 16.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.976912868632199975146607369684, −7.78289653676421805631535892769, −7.37520574778216750961105181761, −6.36053908617028907764290299234, −5.55852010057712242773407238392, −4.83107336339794050725524009083, −4.51025057632657633182941575218, −3.16487212013591831968178080582, −1.60780410794389725190108186443, −0.45151403228862271376371805003, 0.55198531179334388361879941084, 2.30742312885888431569756592018, 3.32760953608167629362486609509, 4.33469512082626019491489188556, 5.23328609753019067256147743596, 5.89853001974281706079860120164, 6.59406733696733477731276896007, 7.41989127685686027076645577333, 7.87687086969086339182354737945, 9.024772461607119401488919657703

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