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Properties

Label 2-2548-91.88-c1-0-0
Degree $2$
Conductor $2548$
Sign $-0.895 + 0.444i$
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  − 1.22·3-s + (−2.34 + 1.35i)5-s − 1.48·9-s − 1.33i·11-s + (2.87 + 2.17i)13-s + (2.87 − 1.66i)15-s + (1.24 + 2.16i)17-s + 4.59i·19-s + (0.104 − 0.180i)23-s + (1.15 − 1.99i)25-s + 5.51·27-s + (−0.639 − 1.10i)29-s + (4.89 + 2.82i)31-s + 1.63i·33-s + (−2.07 − 1.19i)37-s + ⋯
L(s)  = 1  − 0.710·3-s + (−1.04 + 0.604i)5-s − 0.495·9-s − 0.401i·11-s + (0.798 + 0.601i)13-s + (0.743 − 0.429i)15-s + (0.302 + 0.524i)17-s + 1.05i·19-s + (0.0217 − 0.0376i)23-s + (0.230 − 0.399i)25-s + 1.06·27-s + (−0.118 − 0.205i)29-s + (0.878 + 0.507i)31-s + 0.285i·33-s + (−0.340 − 0.196i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $-0.895 + 0.444i$
Analytic conductor: \(20.3458\)
Root analytic conductor: \(4.51064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2548} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2548,\ (\ :1/2),\ -0.895 + 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1442821619\)
\(L(\frac12)\) \(\approx\) \(0.1442821619\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.87 - 2.17i)T \)
good3 \( 1 + 1.22T + 3T^{2} \)
5 \( 1 + (2.34 - 1.35i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + 1.33iT - 11T^{2} \)
17 \( 1 + (-1.24 - 2.16i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 4.59iT - 19T^{2} \)
23 \( 1 + (-0.104 + 0.180i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.639 + 1.10i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.89 - 2.82i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.07 + 1.19i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.55 - 1.47i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.348 - 0.602i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.11 - 1.22i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.09 - 1.89i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.87 - 2.81i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 12.3T + 61T^{2} \)
67 \( 1 + 14.1iT - 67T^{2} \)
71 \( 1 + (1.70 + 0.984i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-10.7 - 6.21i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.16 - 7.21i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.83iT - 83T^{2} \)
89 \( 1 + (10.3 + 5.99i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (14.1 + 8.17i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.284006693116414788431042624209, −8.307115920139795618853190785542, −7.981584807507733617817166982889, −6.90171416319939063586124228240, −6.25310014419668925169272790667, −5.62529946087733080130184534502, −4.53025784412907327827133000467, −3.67673491582578100693322549375, −3.00584300755632241633451782287, −1.43743059282071135201590354740, 0.06576927156018149211050524156, 1.04621279135781085722294237544, 2.71936550248965195477191374573, 3.66460887683461883173863864011, 4.65559364373401351699080665863, 5.19105254793894779425565218883, 6.10259391483382254001648723097, 6.90860427100306668651545486375, 7.79887828377255553760771077880, 8.404396160514558559469077126624

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