↑

Properties

Label 2-75-15.14-c10-0-46
Degree $2$
Conductor $75$
Sign $0.656 - 0.754i$
Analytic cond. $47.6517$
Root an. cond. $6.90302$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 57.7·2-s + (235. + 60.6i)3-s + 2.31e3·4-s + (1.35e4 + 3.50e3i)6-s + 2.27e4i·7-s + 7.44e4·8-s + (5.17e4 + 2.85e4i)9-s + 1.63e5i·11-s + (5.44e5 + 1.40e5i)12-s − 3.76e5i·13-s + 1.31e6i·14-s + 1.93e6·16-s − 1.43e6·17-s + (2.98e6 + 1.64e6i)18-s − 1.20e6·19-s + ⋯
L(s)  = 1  + 1.80·2-s + (0.968 + 0.249i)3-s + 2.25·4-s + (1.74 + 0.450i)6-s + 1.35i·7-s + 2.27·8-s + (0.875 + 0.483i)9-s + 1.01i·11-s + (2.18 + 0.563i)12-s − 1.01i·13-s + 2.44i·14-s + 1.84·16-s − 1.00·17-s + (1.58 + 0.872i)18-s − 0.485·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 - 0.754i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.656 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.656 - 0.754i$
Analytic conductor: \(47.6517\)
Root analytic conductor: \(6.90302\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :5),\ 0.656 - 0.754i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(8.32406 + 3.79262i\)
\(L(\frac12)\) \(\approx\) \(8.32406 + 3.79262i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-235. - 60.6i)T \)
5 \( 1 \)
good2 \( 1 - 57.7T + 1.02e3T^{2} \)
7 \( 1 - 2.27e4iT - 2.82e8T^{2} \)
11 \( 1 - 1.63e5iT - 2.59e10T^{2} \)
13 \( 1 + 3.76e5iT - 1.37e11T^{2} \)
17 \( 1 + 1.43e6T + 2.01e12T^{2} \)
19 \( 1 + 1.20e6T + 6.13e12T^{2} \)
23 \( 1 - 4.68e6T + 4.14e13T^{2} \)
29 \( 1 + 3.09e7iT - 4.20e14T^{2} \)
31 \( 1 - 3.58e7T + 8.19e14T^{2} \)
37 \( 1 + 8.77e7iT - 4.80e15T^{2} \)
41 \( 1 + 6.35e7iT - 1.34e16T^{2} \)
43 \( 1 - 1.28e8iT - 2.16e16T^{2} \)
47 \( 1 - 2.23e8T + 5.25e16T^{2} \)
53 \( 1 - 8.77e6T + 1.74e17T^{2} \)
59 \( 1 + 3.20e8iT - 5.11e17T^{2} \)
61 \( 1 + 3.58e8T + 7.13e17T^{2} \)
67 \( 1 + 2.46e8iT - 1.82e18T^{2} \)
71 \( 1 + 1.70e9iT - 3.25e18T^{2} \)
73 \( 1 + 1.99e9iT - 4.29e18T^{2} \)
79 \( 1 + 1.11e9T + 9.46e18T^{2} \)
83 \( 1 - 3.65e9T + 1.55e19T^{2} \)
89 \( 1 - 5.52e9iT - 3.11e19T^{2} \)
97 \( 1 - 1.19e10iT - 7.37e19T^{2} \)
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

−12.81300884762353005993924249281, −12.06128557348809995800743659491, −10.67365109985841551963480261484, −9.206083573869210265932149380490, −7.80402216169247156101689599095, −6.40886618558387734353653014710, −5.13734360236985262411045783222, −4.15454341175591913784305407900, −2.72967762367858305640680035222, −2.16446323641170235672907842252, 1.29479382271162812675266988899, 2.75111242145121257607509254003, 3.81811443234703007486292269859, 4.63133455174090101561907525137, 6.50001451695584850402570126650, 7.15023306051376676494718218776, 8.693096767446206841811886819680, 10.49195067097694997300616476068, 11.50073730414479264142730769296, 12.82771635284633122372161617495

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