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Properties

Label 2-2548-91.9-c1-0-14
Degree $2$
Conductor $2548$
Sign $0.512 - 0.858i$
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  − 2.30·3-s + (−0.151 + 0.262i)5-s + 2.30·9-s + 6.30·11-s + (1.80 + 3.12i)13-s + (0.348 − 0.603i)15-s + (−0.802 + 1.39i)17-s − 3.69·19-s + (−1 − 1.73i)23-s + (2.45 + 4.25i)25-s + 1.60·27-s + (0.151 − 0.262i)29-s + (−0.197 − 0.341i)31-s − 14.5·33-s + (−4.60 − 7.97i)37-s + ⋯
L(s)  = 1  − 1.32·3-s + (−0.0677 + 0.117i)5-s + 0.767·9-s + 1.90·11-s + (0.499 + 0.866i)13-s + (0.0900 − 0.155i)15-s + (−0.194 + 0.337i)17-s − 0.848·19-s + (−0.208 − 0.361i)23-s + (0.490 + 0.850i)25-s + 0.308·27-s + (0.0281 − 0.0486i)29-s + (−0.0354 − 0.0613i)31-s − 2.52·33-s + (−0.757 − 1.31i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.512 - 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.512 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $0.512 - 0.858i$
Analytic conductor: \(20.3458\)
Root analytic conductor: \(4.51064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2548} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2548,\ (\ :1/2),\ 0.512 - 0.858i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.068634967\)
\(L(\frac12)\) \(\approx\) \(1.068634967\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.80 - 3.12i)T \)
good3 \( 1 + 2.30T + 3T^{2} \)
5 \( 1 + (0.151 - 0.262i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 - 6.30T + 11T^{2} \)
17 \( 1 + (0.802 - 1.39i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 3.69T + 19T^{2} \)
23 \( 1 + (1 + 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.151 + 0.262i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.197 + 0.341i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.60 + 7.97i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.697 - 1.20i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.95 - 5.11i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.80 + 10.0i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.40 + 9.36i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.80 - 4.85i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 7.21T + 61T^{2} \)
67 \( 1 - 13T + 67T^{2} \)
71 \( 1 + (-4.90 - 8.50i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.60 - 9.70i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.39T + 83T^{2} \)
89 \( 1 + (-2.54 - 4.40i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.04 + 5.27i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.017806848385753300707925895852, −8.448304754713602222168832377472, −7.05960834470264692806277238911, −6.62464715349105849162529157504, −6.09228870963960027675404486176, −5.20442026681040807033728457897, −4.21117289506574890939058776274, −3.71078670557848800428882458525, −2.00844561655192492590372530620, −0.970320479531943023578825944872, 0.56619345701081749679986706381, 1.56914379150083745464011073492, 3.14543993601826476214973553723, 4.18847659397655612075307156502, 4.83192574044529613217456658958, 5.85977393884859305367792927572, 6.35009618040281575781826444118, 6.91361161938621826348121664451, 8.030633164874892414686015692795, 8.847697037049990428326510180455

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