↑

Properties

Label 2-375-25.11-c1-0-5
Degree $2$
Conductor $375$
Sign $0.344 - 0.938i$
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  + (−1.38 + 1.00i)2-s + (−0.309 + 0.951i)3-s + (0.281 − 0.867i)4-s + (−0.527 − 1.62i)6-s + 3.94·7-s + (−0.573 − 1.76i)8-s + (−0.809 − 0.587i)9-s + (4.78 − 3.47i)11-s + (0.737 + 0.535i)12-s + (2.66 + 1.93i)13-s + (−5.44 + 3.95i)14-s + (4.03 + 2.93i)16-s + (−0.836 − 2.57i)17-s + 1.70·18-s + (−0.728 − 2.24i)19-s + ⋯
L(s)  = 1  + (−0.976 + 0.709i)2-s + (−0.178 + 0.549i)3-s + (0.140 − 0.433i)4-s + (−0.215 − 0.662i)6-s + 1.49·7-s + (−0.202 − 0.624i)8-s + (−0.269 − 0.195i)9-s + (1.44 − 1.04i)11-s + (0.212 + 0.154i)12-s + (0.738 + 0.536i)13-s + (−1.45 + 1.05i)14-s + (1.00 + 0.733i)16-s + (−0.202 − 0.624i)17-s + 0.402·18-s + (−0.167 − 0.514i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.344 - 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.344 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $0.344 - 0.938i$
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: \((2,\ 375,\ (\ :1/2),\ 0.344 - 0.938i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.772259 + 0.539240i\)
\(L(\frac12)\) \(\approx\) \(0.772259 + 0.539240i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.38 - 1.00i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 - 3.94T + 7T^{2} \)
11 \( 1 + (-4.78 + 3.47i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-2.66 - 1.93i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.836 + 2.57i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.728 + 2.24i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.472 - 0.343i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (1.20 - 3.72i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.837 - 2.57i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (0.0168 + 0.0122i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (1.19 + 0.865i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 1.27T + 43T^{2} \)
47 \( 1 + (1.67 - 5.16i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (0.870 - 2.67i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-3.79 - 2.75i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (4.51 - 3.28i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (1.86 + 5.73i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (2.50 - 7.70i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-10.7 + 7.82i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-5.14 + 15.8i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.241 + 0.743i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-2.80 + 2.04i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-0.758 + 2.33i)T + (-78.4 - 57.0i)T^{2} \)
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

−11.33002014850668478788161507885, −10.65738380037419575243496320456, −9.232177112991334886179495659834, −8.875337554795964543493206837785, −8.063678636074351297307890526710, −6.92074521476603065014123750637, −6.02504053883970364598613316144, −4.69213924341779913792347105502, −3.59593659762273429428076372639, −1.24229444944271261465591660443, 1.29728451353331249801466750643, 2.04063584341240218891677689562, 4.05809881548241987104843882082, 5.37038647393993493425842138095, 6.56428116469734175851869716128, 7.892716723715126845701137572990, 8.385137105860356243938368676786, 9.395588177566629892500695043825, 10.37176700842121055547008017874, 11.24705983540410762669513899140

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