Properties

Label 2-75-5.3-c4-0-6
Degree $2$
Conductor $75$
Sign $-0.326 - 0.945i$
Analytic cond. $7.75274$
Root an. cond. $2.78437$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−5.44 + 5.44i)2-s + (3.67 + 3.67i)3-s − 43.3i·4-s − 40.0·6-s + (19.2 − 19.2i)7-s + (149. + 149. i)8-s + 27i·9-s + 185.·11-s + (159. − 159. i)12-s + (48.5 + 48.5i)13-s + 209. i·14-s − 932.·16-s + (−147. + 147. i)17-s + (−147. − 147. i)18-s − 140. i·19-s + ⋯
L(s)  = 1  + (−1.36 + 1.36i)2-s + (0.408 + 0.408i)3-s − 2.71i·4-s − 1.11·6-s + (0.392 − 0.392i)7-s + (2.33 + 2.33i)8-s + 0.333i·9-s + 1.52·11-s + (1.10 − 1.10i)12-s + (0.287 + 0.287i)13-s + 1.06i·14-s − 3.64·16-s + (−0.510 + 0.510i)17-s + (−0.454 − 0.454i)18-s − 0.388i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.326 - 0.945i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.326 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.326 - 0.945i$
Analytic conductor: \(7.75274\)
Root analytic conductor: \(2.78437\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :2),\ -0.326 - 0.945i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.597837 + 0.838927i\)
\(L(\frac12)\) \(\approx\) \(0.597837 + 0.838927i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.67 - 3.67i)T \)
5 \( 1 \)
good2 \( 1 + (5.44 - 5.44i)T - 16iT^{2} \)
7 \( 1 + (-19.2 + 19.2i)T - 2.40e3iT^{2} \)
11 \( 1 - 185.T + 1.46e4T^{2} \)
13 \( 1 + (-48.5 - 48.5i)T + 2.85e4iT^{2} \)
17 \( 1 + (147. - 147. i)T - 8.35e4iT^{2} \)
19 \( 1 + 140. iT - 1.30e5T^{2} \)
23 \( 1 + (-177. - 177. i)T + 2.79e5iT^{2} \)
29 \( 1 - 588. iT - 7.07e5T^{2} \)
31 \( 1 - 1.41e3T + 9.23e5T^{2} \)
37 \( 1 + (763. - 763. i)T - 1.87e6iT^{2} \)
41 \( 1 - 995.T + 2.82e6T^{2} \)
43 \( 1 + (-657. - 657. i)T + 3.41e6iT^{2} \)
47 \( 1 + (-102. + 102. i)T - 4.87e6iT^{2} \)
53 \( 1 + (-365. - 365. i)T + 7.89e6iT^{2} \)
59 \( 1 - 5.80e3iT - 1.21e7T^{2} \)
61 \( 1 - 6.28e3T + 1.38e7T^{2} \)
67 \( 1 + (-5.49e3 + 5.49e3i)T - 2.01e7iT^{2} \)
71 \( 1 + 7.66e3T + 2.54e7T^{2} \)
73 \( 1 + (6.40e3 + 6.40e3i)T + 2.83e7iT^{2} \)
79 \( 1 + 5.69e3iT - 3.89e7T^{2} \)
83 \( 1 + (999. + 999. i)T + 4.74e7iT^{2} \)
89 \( 1 + 3.09e3iT - 6.27e7T^{2} \)
97 \( 1 + (-5.03e3 + 5.03e3i)T - 8.85e7iT^{2} \)
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

−14.53100967119945562922781022514, −13.74167289340980353455233323065, −11.39837382665745534721297686879, −10.31650452641542344300996074503, −9.190149984781258471495861794573, −8.537336106092434049922447047236, −7.24287805417863313466005197587, −6.21878261684051567398131830055, −4.51240597080910658627572458635, −1.31101374695103413165768420687, 0.999655996067606689903239902481, 2.37989792492852585213167979234, 3.91748909226179845771219584062, 6.85115243911160431006956564254, 8.230277897450109010067023540050, 8.961837451665461991740794590060, 9.964095571279687791110569319213, 11.35006549393439243708039721553, 11.95267379969494159762233473986, 13.02301726954245399482626051217

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