Properties

Label 2-75-25.13-c2-0-7
Degree $2$
Conductor $75$
Sign $0.904 + 0.426i$
Analytic cond. $2.04360$
Root an. cond. $1.42954$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.92 − 0.463i)2-s + (0.786 − 1.54i)3-s + (4.54 − 1.47i)4-s + (−2.61 + 4.26i)5-s + (1.58 − 4.88i)6-s + (2.66 − 2.66i)7-s + (2.07 − 1.05i)8-s + (−1.76 − 2.42i)9-s + (−5.67 + 13.6i)10-s + (−2.59 − 1.88i)11-s + (1.29 − 8.18i)12-s + (−14.9 − 2.36i)13-s + (6.57 − 9.04i)14-s + (4.52 + 7.38i)15-s + (−9.90 + 7.19i)16-s + (5.47 + 10.7i)17-s + ⋯
L(s)  = 1  + (1.46 − 0.231i)2-s + (0.262 − 0.514i)3-s + (1.13 − 0.369i)4-s + (−0.522 + 0.852i)5-s + (0.264 − 0.813i)6-s + (0.381 − 0.381i)7-s + (0.258 − 0.131i)8-s + (−0.195 − 0.269i)9-s + (−0.567 + 1.36i)10-s + (−0.235 − 0.171i)11-s + (0.108 − 0.682i)12-s + (−1.15 − 0.182i)13-s + (0.469 − 0.646i)14-s + (0.301 + 0.492i)15-s + (−0.619 + 0.449i)16-s + (0.322 + 0.632i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.904 + 0.426i$
Analytic conductor: \(2.04360\)
Root analytic conductor: \(1.42954\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :1),\ 0.904 + 0.426i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.36476 - 0.529769i\)
\(L(\frac12)\) \(\approx\) \(2.36476 - 0.529769i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.786 + 1.54i)T \)
5 \( 1 + (2.61 - 4.26i)T \)
good2 \( 1 + (-2.92 + 0.463i)T + (3.80 - 1.23i)T^{2} \)
7 \( 1 + (-2.66 + 2.66i)T - 49iT^{2} \)
11 \( 1 + (2.59 + 1.88i)T + (37.3 + 115. i)T^{2} \)
13 \( 1 + (14.9 + 2.36i)T + (160. + 52.2i)T^{2} \)
17 \( 1 + (-5.47 - 10.7i)T + (-169. + 233. i)T^{2} \)
19 \( 1 + (-22.7 - 7.38i)T + (292. + 212. i)T^{2} \)
23 \( 1 + (-1.40 - 8.87i)T + (-503. + 163. i)T^{2} \)
29 \( 1 + (-30.5 + 9.93i)T + (680. - 494. i)T^{2} \)
31 \( 1 + (-11.6 + 35.8i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (-10.4 + 66.1i)T + (-1.30e3 - 423. i)T^{2} \)
41 \( 1 + (55.8 - 40.5i)T + (519. - 1.59e3i)T^{2} \)
43 \( 1 + (34.9 + 34.9i)T + 1.84e3iT^{2} \)
47 \( 1 + (-35.6 - 18.1i)T + (1.29e3 + 1.78e3i)T^{2} \)
53 \( 1 + (-33.9 + 66.6i)T + (-1.65e3 - 2.27e3i)T^{2} \)
59 \( 1 + (-6.33 - 8.72i)T + (-1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (-65.9 - 47.9i)T + (1.14e3 + 3.53e3i)T^{2} \)
67 \( 1 + (-3.44 - 6.76i)T + (-2.63e3 + 3.63e3i)T^{2} \)
71 \( 1 + (-30.8 - 94.9i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (3.29 + 20.7i)T + (-5.06e3 + 1.64e3i)T^{2} \)
79 \( 1 + (116. - 37.9i)T + (5.04e3 - 3.66e3i)T^{2} \)
83 \( 1 + (69.6 - 35.4i)T + (4.04e3 - 5.57e3i)T^{2} \)
89 \( 1 + (29.4 - 40.5i)T + (-2.44e3 - 7.53e3i)T^{2} \)
97 \( 1 + (-53.7 - 27.3i)T + (5.53e3 + 7.61e3i)T^{2} \)
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

−14.22994560247720291614039960381, −13.26771968342922504160860258636, −12.12344694172349350117688149642, −11.44662567328992685806415113024, −10.09900630990066770865967252955, −8.012024301958930337323246064511, −6.97543335057436400214831050238, −5.53062023093666039052422074261, −3.95291985412912090449165374440, −2.65581179470118868928378026408, 3.06015781994512482216657270468, 4.74581160231867603173938160936, 5.15636373640385393640575532854, 7.12341739802577109702178035948, 8.572528560894012925072147018662, 9.866504179218571376870899553909, 11.76229655583422183693976426633, 12.17157538678531817112704144683, 13.46406536768718178460830742342, 14.34483373112197257071902089643

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