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Properties

Label 2-2548-7.4-c1-0-37
Degree $2$
Conductor $2548$
Sign $-0.991 + 0.126i$
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  + (1.71 − 2.97i)3-s + (0.459 + 0.795i)5-s + (−4.39 − 7.60i)9-s + (−0.802 + 1.38i)11-s + 13-s + 3.15·15-s + (−1.36 + 2.35i)17-s + (−1.05 − 1.83i)19-s + (−2.71 − 4.70i)23-s + (2.07 − 3.59i)25-s − 19.8·27-s − 6.50·29-s + (2.83 − 4.90i)31-s + (2.75 + 4.76i)33-s + (−4.71 − 8.16i)37-s + ⋯
L(s)  = 1  + (0.990 − 1.71i)3-s + (0.205 + 0.355i)5-s + (−1.46 − 2.53i)9-s + (−0.241 + 0.418i)11-s + 0.277·13-s + 0.814·15-s + (−0.330 + 0.571i)17-s + (−0.242 − 0.420i)19-s + (−0.566 − 0.981i)23-s + (0.415 − 0.719i)25-s − 3.82·27-s − 1.20·29-s + (0.509 − 0.881i)31-s + (0.479 + 0.830i)33-s + (−0.775 − 1.34i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $-0.991 + 0.126i$
Analytic conductor: \(20.3458\)
Root analytic conductor: \(4.51064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2548} (1145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2548,\ (\ :1/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.851379830\)
\(L(\frac12)\) \(\approx\) \(1.851379830\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good3 \( 1 + (-1.71 + 2.97i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.459 - 0.795i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.802 - 1.38i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.36 - 2.35i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.05 + 1.83i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.71 + 4.70i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.50T + 29T^{2} \)
31 \( 1 + (-2.83 + 4.90i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.71 + 8.16i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.35T + 41T^{2} \)
43 \( 1 + 0.513T + 43T^{2} \)
47 \( 1 + (-2.31 - 4.01i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.81 - 4.87i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.66 + 6.35i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.303 + 0.525i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.72 + 9.91i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 15.3T + 71T^{2} \)
73 \( 1 + (5.09 - 8.81i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.26 + 7.38i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 13.0T + 83T^{2} \)
89 \( 1 + (6.26 + 10.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 7.21T + 97T^{2} \)
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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

−8.493908286359937397669121686378, −7.64873403016629711101940794245, −7.18186189060067870620732257861, −6.30470578874867318490149590904, −5.92433254759216094930563745905, −4.35373296869408667138709882592, −3.34239639308767275478762647485, −2.36473968577114826065690033881, −1.90171229427781406769804936287, −0.48779948226781878516914523499, 1.82020205940817116935194548616, 2.98364858458576681767738987899, 3.58416118141684586700075081828, 4.43400754846009872071625912236, 5.22187367727928112771532075269, 5.76711404683086004591544156017, 7.18924434313893226456777453268, 8.110815298754774586008835828554, 8.684732686609733270438054620889, 9.225201813319451556653362958623

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