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Properties

Label 2-2548-91.23-c1-0-10
Degree $2$
Conductor $2548$
Sign $-0.977 - 0.211i$
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  + (−0.450 + 0.780i)3-s + (1.46 + 0.844i)5-s + (1.09 + 1.89i)9-s + (−1.07 − 0.620i)11-s + (3.32 + 1.40i)13-s + (−1.31 + 0.760i)15-s − 3.60·17-s + (−6.45 + 3.72i)19-s − 8.37·23-s + (−1.07 − 1.86i)25-s − 4.67·27-s + (3.09 + 5.35i)29-s + (−1.13 + 0.655i)31-s + (0.967 − 0.558i)33-s − 10.2i·37-s + ⋯
L(s)  = 1  + (−0.260 + 0.450i)3-s + (0.654 + 0.377i)5-s + (0.364 + 0.631i)9-s + (−0.323 − 0.186i)11-s + (0.921 + 0.389i)13-s + (−0.340 + 0.196i)15-s − 0.874·17-s + (−1.47 + 0.854i)19-s − 1.74·23-s + (−0.214 − 0.372i)25-s − 0.899·27-s + (0.574 + 0.994i)29-s + (−0.203 + 0.117i)31-s + (0.168 − 0.0972i)33-s − 1.69i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.211i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 - 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $-0.977 - 0.211i$
Analytic conductor: \(20.3458\)
Root analytic conductor: \(4.51064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2548} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2548,\ (\ :1/2),\ -0.977 - 0.211i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9327276079\)
\(L(\frac12)\) \(\approx\) \(0.9327276079\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.32 - 1.40i)T \)
good3 \( 1 + (0.450 - 0.780i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.46 - 0.844i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.07 + 0.620i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 3.60T + 17T^{2} \)
19 \( 1 + (6.45 - 3.72i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 8.37T + 23T^{2} \)
29 \( 1 + (-3.09 - 5.35i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.13 - 0.655i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 10.2iT - 37T^{2} \)
41 \( 1 + (-1.40 + 0.812i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.78 - 8.27i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-10.3 - 5.97i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.95 + 8.58i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 4.74iT - 59T^{2} \)
61 \( 1 + (-6.19 - 10.7i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.69 + 2.71i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.90 - 2.25i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.42 - 2.55i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.11 + 5.38i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 17.4iT - 83T^{2} \)
89 \( 1 + 5.86iT - 89T^{2} \)
97 \( 1 + (1.64 + 0.951i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.335760830588410293152148671782, −8.463005304758518824471165104121, −7.87882928704762404548155293323, −6.73884063472153379414468968900, −6.13195960989922727899692135957, −5.49552416489563316589205874310, −4.34283297860881042811564591349, −3.89239718013481213206270895880, −2.41768227279678939508641192708, −1.74955218879433957249284150024, 0.29398279037947903660177746714, 1.60813246845014289711587532352, 2.43017436454205568481054230688, 3.82568750382974604526491341892, 4.52209538789080589432937291191, 5.63739586183881926877910981676, 6.27871139826992079026998846512, 6.76854291676603911850310501328, 7.85062470007706336456387457166, 8.573214275696327918543119716786

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