Properties

Label 2-375-25.11-c1-0-3
Degree $2$
Conductor $375$
Sign $-0.175 - 0.984i$
Analytic cond. $2.99439$
Root an. cond. $1.73043$
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  + (0.0726 − 0.0527i)2-s + (−0.309 + 0.951i)3-s + (−0.615 + 1.89i)4-s + (0.0277 + 0.0854i)6-s + 4.36·7-s + (0.110 + 0.341i)8-s + (−0.809 − 0.587i)9-s + (−3.55 + 2.58i)11-s + (−1.61 − 1.17i)12-s + (1.60 + 1.16i)13-s + (0.316 − 0.230i)14-s + (−3.19 − 2.32i)16-s + (0.308 + 0.948i)17-s − 0.0898·18-s + (0.417 + 1.28i)19-s + ⋯
L(s)  = 1  + (0.0513 − 0.0373i)2-s + (−0.178 + 0.549i)3-s + (−0.307 + 0.947i)4-s + (0.0113 + 0.0348i)6-s + 1.64·7-s + (0.0391 + 0.120i)8-s + (−0.269 − 0.195i)9-s + (−1.07 + 0.778i)11-s + (−0.465 − 0.337i)12-s + (0.444 + 0.323i)13-s + (0.0846 − 0.0615i)14-s + (−0.799 − 0.580i)16-s + (0.0747 + 0.229i)17-s − 0.0211·18-s + (0.0957 + 0.294i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.175 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $-0.175 - 0.984i$
Analytic conductor: \(2.99439\)
Root analytic conductor: \(1.73043\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{375} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 375,\ (\ :1/2),\ -0.175 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.824950 + 0.985399i\)
\(L(\frac12)\) \(\approx\) \(0.824950 + 0.985399i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 \)
good2 \( 1 + (-0.0726 + 0.0527i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 - 4.36T + 7T^{2} \)
11 \( 1 + (3.55 - 2.58i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-1.60 - 1.16i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.308 - 0.948i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-0.417 - 1.28i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (1.90 - 1.38i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (2.46 - 7.58i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.13 + 3.49i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-1.16 - 0.844i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-4.83 - 3.51i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 2.68T + 43T^{2} \)
47 \( 1 + (-3.38 + 10.4i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-3.41 + 10.5i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (5.41 + 3.93i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-7.64 + 5.55i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-3.99 - 12.2i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-2.26 + 6.96i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-0.343 + 0.249i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.96 - 6.04i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.227 - 0.700i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-7.91 + 5.75i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-0.0104 + 0.0320i)T + (-78.4 - 57.0i)T^{2} \)
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

−11.51229981519572558281364957470, −10.91487134078104525332162039980, −9.836180415309668623758228082491, −8.677075757847279903058430451036, −8.017380832665876833073059202560, −7.21081957935003572640959520036, −5.43882852891471522368544453897, −4.69483238893983754585536512176, −3.73029537537692144330059309782, −2.11281045513025671116268783912, 0.937509892255330249714474915915, 2.34930471001251065910968752903, 4.40039514692223320373617349121, 5.40181206180088925081757646444, 6.01011094829085546039094818703, 7.53815329286553843325541444641, 8.205600497612489917701102959884, 9.178567701448891942887944361588, 10.62058776980871532289914885041, 10.92407070709835699419712558363

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