↑

Properties

Label 2-2548-91.88-c1-0-27
Degree $2$
Conductor $2548$
Sign $0.845 - 0.533i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.98·3-s + (0.626 − 0.361i)5-s + 5.89·9-s + 2.58i·11-s + (0.896 + 3.49i)13-s + (1.86 − 1.07i)15-s + (−2.26 − 3.93i)17-s + 7.22i·19-s + (−0.0997 + 0.172i)23-s + (−2.23 + 3.87i)25-s + 8.64·27-s + (2.84 + 4.93i)29-s + (5.36 + 3.09i)31-s + 7.70i·33-s + (−4.01 − 2.32i)37-s + ⋯
L(s)  = 1  + 1.72·3-s + (0.280 − 0.161i)5-s + 1.96·9-s + 0.779i·11-s + (0.248 + 0.968i)13-s + (0.482 − 0.278i)15-s + (−0.550 − 0.953i)17-s + 1.65i·19-s + (−0.0208 + 0.0360i)23-s + (−0.447 + 0.775i)25-s + 1.66·27-s + (0.529 + 0.916i)29-s + (0.963 + 0.556i)31-s + 1.34i·33-s + (−0.660 − 0.381i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 - 0.533i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.845 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $0.845 - 0.533i$
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: \((2,\ 2548,\ (\ :1/2),\ 0.845 - 0.533i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.660777135\)
\(L(\frac12)\) \(\approx\) \(3.660777135\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.896 - 3.49i)T \)
good3 \( 1 - 2.98T + 3T^{2} \)
5 \( 1 + (-0.626 + 0.361i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 - 2.58iT - 11T^{2} \)
17 \( 1 + (2.26 + 3.93i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 7.22iT - 19T^{2} \)
23 \( 1 + (0.0997 - 0.172i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.84 - 4.93i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.36 - 3.09i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.01 + 2.32i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-8.26 + 4.77i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.54 + 6.14i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.838 + 0.484i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.13 + 3.69i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.48 + 4.32i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 8.73T + 61T^{2} \)
67 \( 1 + 2.73iT - 67T^{2} \)
71 \( 1 + (-0.0784 - 0.0453i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.56 + 2.05i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.96 + 6.86i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.65iT - 83T^{2} \)
89 \( 1 + (12.2 + 7.06i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.09 - 0.630i)T + (48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.838261380483597109727043110044, −8.472497042167133697265518993057, −7.33810788886002947577920602789, −7.13975639960289922225137269480, −5.92346157382717000358004135428, −4.75635101646456839669547996740, −4.01547463391556352978777853192, −3.22014007626498053791280722523, −2.17470728437372467708357205802, −1.57257452655174935348314333156, 1.02535750293550658023507396986, 2.55573076290977351127214317372, 2.72862689368366647366409812724, 3.88209192911069144222479127612, 4.56389463314289890341200807458, 5.90189019590505853363567565196, 6.57985195486832597177844471683, 7.62243660169443712818100252285, 8.217605902221525229591244669191, 8.684137292843395640364953821164

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