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Properties

Label 2-2548-91.81-c1-0-31
Degree $2$
Conductor $2548$
Sign $0.943 - 0.331i$
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  + 3.21·3-s + (−0.0383 − 0.0664i)5-s + 7.36·9-s − 2.11·11-s + (1.62 + 3.21i)13-s + (−0.123 − 0.214i)15-s + (3.61 + 6.25i)17-s + 0.676·19-s + (1.83 − 3.18i)23-s + (2.49 − 4.32i)25-s + 14.0·27-s + (−1.90 − 3.30i)29-s + (−2.07 + 3.60i)31-s − 6.82·33-s + (−4.05 + 7.01i)37-s + ⋯
L(s)  = 1  + 1.85·3-s + (−0.0171 − 0.0297i)5-s + 2.45·9-s − 0.639·11-s + (0.450 + 0.892i)13-s + (−0.0319 − 0.0552i)15-s + (0.876 + 1.51i)17-s + 0.155·19-s + (0.383 − 0.663i)23-s + (0.499 − 0.865i)25-s + 2.70·27-s + (−0.354 − 0.614i)29-s + (−0.373 + 0.646i)31-s − 1.18·33-s + (−0.666 + 1.15i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.331i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 - 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $0.943 - 0.331i$
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: \((2,\ 2548,\ (\ :1/2),\ 0.943 - 0.331i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.786349647\)
\(L(\frac12)\) \(\approx\) \(3.786349647\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 - 3.21i)T \)
good3 \( 1 - 3.21T + 3T^{2} \)
5 \( 1 + (0.0383 + 0.0664i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 2.11T + 11T^{2} \)
17 \( 1 + (-3.61 - 6.25i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 0.676T + 19T^{2} \)
23 \( 1 + (-1.83 + 3.18i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.90 + 3.30i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.07 - 3.60i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.05 - 7.01i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.01 + 8.69i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.88 + 5.00i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.43 - 11.1i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.45 + 4.25i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.58 - 4.47i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 7.36T + 61T^{2} \)
67 \( 1 + 7.79T + 67T^{2} \)
71 \( 1 + (-0.808 + 1.40i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.339 + 0.587i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.30 - 2.25i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.48T + 83T^{2} \)
89 \( 1 + (1.55 - 2.69i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.11 + 10.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.655524325096311330028090814078, −8.448328805837601563269496065907, −7.59578219729258835894283456123, −6.92157561343974345054228792439, −5.94755979507289716802980272121, −4.68451528788326288243770111729, −3.91205265940984148519095004359, −3.20867764002852468198760273259, −2.28142188230780146951516962478, −1.42679765838226883022973741202, 1.13164257338986033208208745816, 2.34234827248845606794797149211, 3.19298386644473198696155474503, 3.56301531964479120104450250516, 4.86443946292467940887734736214, 5.57567302001383743064692461112, 7.01923684149899862899247620010, 7.56618333772780173494653043637, 7.989497885518663197417590432804, 8.998762756221520284178649135474

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