Properties

Label 2-75-15.8-c3-0-1
Degree $2$
Conductor $75$
Sign $0.329 - 0.944i$
Analytic cond. $4.42514$
Root an. cond. $2.10360$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.39 − 1.39i)2-s + (−4.93 + 1.62i)3-s − 4.10i·4-s + (9.15 + 4.61i)6-s + (−3.80 + 3.80i)7-s + (−16.8 + 16.8i)8-s + (21.7 − 16.0i)9-s + 61.8i·11-s + (6.67 + 20.2i)12-s + (48.1 + 48.1i)13-s + 10.6·14-s + 14.3·16-s + (−47.5 − 47.5i)17-s + (−52.7 − 7.89i)18-s + 93.4i·19-s + ⋯
L(s)  = 1  + (−0.493 − 0.493i)2-s + (−0.949 + 0.312i)3-s − 0.513i·4-s + (0.623 + 0.314i)6-s + (−0.205 + 0.205i)7-s + (−0.746 + 0.746i)8-s + (0.804 − 0.594i)9-s + 1.69i·11-s + (0.160 + 0.487i)12-s + (1.02 + 1.02i)13-s + 0.202·14-s + 0.223·16-s + (−0.679 − 0.679i)17-s + (−0.690 − 0.103i)18-s + 1.12i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.329 - 0.944i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.329 - 0.944i$
Analytic conductor: \(4.42514\)
Root analytic conductor: \(2.10360\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :3/2),\ 0.329 - 0.944i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.434481 + 0.308505i\)
\(L(\frac12)\) \(\approx\) \(0.434481 + 0.308505i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.93 - 1.62i)T \)
5 \( 1 \)
good2 \( 1 + (1.39 + 1.39i)T + 8iT^{2} \)
7 \( 1 + (3.80 - 3.80i)T - 343iT^{2} \)
11 \( 1 - 61.8iT - 1.33e3T^{2} \)
13 \( 1 + (-48.1 - 48.1i)T + 2.19e3iT^{2} \)
17 \( 1 + (47.5 + 47.5i)T + 4.91e3iT^{2} \)
19 \( 1 - 93.4iT - 6.85e3T^{2} \)
23 \( 1 + (-33.7 + 33.7i)T - 1.21e4iT^{2} \)
29 \( 1 + 179.T + 2.43e4T^{2} \)
31 \( 1 + 123.T + 2.97e4T^{2} \)
37 \( 1 + (10.5 - 10.5i)T - 5.06e4iT^{2} \)
41 \( 1 + 61.8iT - 6.89e4T^{2} \)
43 \( 1 + (-133. - 133. i)T + 7.95e4iT^{2} \)
47 \( 1 + (56.9 + 56.9i)T + 1.03e5iT^{2} \)
53 \( 1 + (234. - 234. i)T - 1.48e5iT^{2} \)
59 \( 1 - 260.T + 2.05e5T^{2} \)
61 \( 1 - 240.T + 2.26e5T^{2} \)
67 \( 1 + (320. - 320. i)T - 3.00e5iT^{2} \)
71 \( 1 - 1.08e3iT - 3.57e5T^{2} \)
73 \( 1 + (208. + 208. i)T + 3.89e5iT^{2} \)
79 \( 1 + 676. iT - 4.93e5T^{2} \)
83 \( 1 + (71.3 - 71.3i)T - 5.71e5iT^{2} \)
89 \( 1 + 228.T + 7.04e5T^{2} \)
97 \( 1 + (89.5 - 89.5i)T - 9.12e5iT^{2} \)
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

−14.43890426113281236840817410007, −12.80845896842638973433841290514, −11.75907110496969087104942938244, −10.92449242034606585092076075831, −9.844737490939467756999771295320, −9.106984417863135578948082458199, −6.99674042041571388352783945307, −5.79071503707688450726568997723, −4.37724228959192911326233579985, −1.70652528299794797529009203634, 0.45848591897039522046594669914, 3.55818955070879909799372827467, 5.68935315206487167459074278691, 6.67705465403284067806262177599, 7.959738906479968873934924772350, 8.994203020063630611529696408757, 10.74330610202911279360159508694, 11.42087781834644556750464165393, 12.97125083194323158492552778566, 13.35933076925716069309239556113

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