↑

Properties

Label 2-75-25.21-c1-0-3
Degree $2$
Conductor $75$
Sign $0.535 + 0.844i$
Analytic cond. $0.598878$
Root an. cond. $0.773872$
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  + (0.309 − 0.951i)2-s + (−0.809 − 0.587i)3-s + (0.809 + 0.587i)4-s + (1.80 − 1.31i)5-s + (−0.809 + 0.587i)6-s − 4.47·7-s + (2.42 − 1.76i)8-s + (0.309 + 0.951i)9-s + (−0.690 − 2.12i)10-s + (−0.381 + 1.17i)11-s + (−0.309 − 0.951i)12-s + (1.73 + 5.34i)13-s + (−1.38 + 4.25i)14-s − 2.23·15-s + (−0.309 − 0.951i)16-s + (−3.11 + 2.26i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (−0.467 − 0.339i)3-s + (0.404 + 0.293i)4-s + (0.809 − 0.587i)5-s + (−0.330 + 0.239i)6-s − 1.69·7-s + (0.858 − 0.623i)8-s + (0.103 + 0.317i)9-s + (−0.218 − 0.672i)10-s + (−0.115 + 0.354i)11-s + (−0.0892 − 0.274i)12-s + (0.481 + 1.48i)13-s + (−0.369 + 1.13i)14-s − 0.577·15-s + (−0.0772 − 0.237i)16-s + (−0.756 + 0.549i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.535 + 0.844i$
Analytic conductor: \(0.598878\)
Root analytic conductor: \(0.773872\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :1/2),\ 0.535 + 0.844i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.887884 - 0.488118i\)
\(L(\frac12)\) \(\approx\) \(0.887884 - 0.488118i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (-1.80 + 1.31i)T \)
good2 \( 1 + (-0.309 + 0.951i)T + (-1.61 - 1.17i)T^{2} \)
7 \( 1 + 4.47T + 7T^{2} \)
11 \( 1 + (0.381 - 1.17i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-1.73 - 5.34i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (3.11 - 2.26i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1 - 0.726i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-1.38 + 4.25i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (5.35 + 3.88i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (2.23 - 1.62i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.954 + 2.93i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.11 + 3.44i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 7.70T + 43T^{2} \)
47 \( 1 + (-0.618 - 0.449i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.92 + 2.12i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.23 + 3.80i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-0.5 + 1.53i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (0.618 - 0.449i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (-4.23 - 3.07i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.5 - 7.69i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (10.0 - 7.33i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-1.66 + 5.11i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (1.73 + 1.26i)T + (29.9 + 92.2i)T^{2} \)
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

−13.84782512551672825677613236966, −12.84970055048529302820432863721, −12.55632039327822063579897882799, −11.18523460698977763483873020426, −10.05608923728147184240434983330, −8.993629421739908416521117933153, −6.93577447550883033588940439399, −6.15541113893471457139290388990, −4.10086683798978184237670281144, −2.14907754764985838790991192471, 3.08850976434554025446558559904, 5.52587764509741470347026758162, 6.21666430712940439343330207693, 7.23810600244196575009582988232, 9.311649588162327759794256093831, 10.35665720755647499574623760551, 11.10023413838207734059851263087, 12.92972869034885959123359738147, 13.61756424588925591930250261797, 15.05616389358667091129833791522

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