↑

Properties

Label 2-2475-2475.1913-c0-0-1
Degree $2$
Conductor $2475$
Sign $-0.889 - 0.457i$
Analytic cond. $1.23518$
Root an. cond. $1.11138$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.406 − 0.913i)3-s + (0.207 − 0.978i)4-s − i·5-s + (−0.669 + 0.743i)9-s + (−0.994 − 0.104i)11-s + (−0.978 + 0.207i)12-s + (−0.913 + 0.406i)15-s + (−0.913 − 0.406i)16-s + (−0.978 − 0.207i)20-s + (−0.638 + 1.66i)23-s − 25-s + (0.951 + 0.309i)27-s + (−1.33 − 1.47i)31-s + (0.309 + 0.951i)33-s + (0.587 + 0.809i)36-s + (0.302 − 1.90i)37-s + ⋯
L(s)  = 1  + (−0.406 − 0.913i)3-s + (0.207 − 0.978i)4-s − i·5-s + (−0.669 + 0.743i)9-s + (−0.994 − 0.104i)11-s + (−0.978 + 0.207i)12-s + (−0.913 + 0.406i)15-s + (−0.913 − 0.406i)16-s + (−0.978 − 0.207i)20-s + (−0.638 + 1.66i)23-s − 25-s + (0.951 + 0.309i)27-s + (−1.33 − 1.47i)31-s + (0.309 + 0.951i)33-s + (0.587 + 0.809i)36-s + (0.302 − 1.90i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.889 - 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.889 - 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $-0.889 - 0.457i$
Analytic conductor: \(1.23518\)
Root analytic conductor: \(1.11138\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (1913, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2475,\ (\ :0),\ -0.889 - 0.457i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6334936614\)
\(L(\frac12)\) \(\approx\) \(0.6334936614\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.406 + 0.913i)T \)
5 \( 1 + iT \)
11 \( 1 + (0.994 + 0.104i)T \)
good2 \( 1 + (-0.207 + 0.978i)T^{2} \)
7 \( 1 + (-0.866 - 0.5i)T^{2} \)
13 \( 1 + (0.207 + 0.978i)T^{2} \)
17 \( 1 + (-0.587 + 0.809i)T^{2} \)
19 \( 1 + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + (0.638 - 1.66i)T + (-0.743 - 0.669i)T^{2} \)
29 \( 1 + (0.104 + 0.994i)T^{2} \)
31 \( 1 + (1.33 + 1.47i)T + (-0.104 + 0.994i)T^{2} \)
37 \( 1 + (-0.302 + 1.90i)T + (-0.951 - 0.309i)T^{2} \)
41 \( 1 + (-0.978 + 0.207i)T^{2} \)
43 \( 1 + (0.866 + 0.5i)T^{2} \)
47 \( 1 + (0.516 - 0.0270i)T + (0.994 - 0.104i)T^{2} \)
53 \( 1 + (-1.12 - 0.571i)T + (0.587 + 0.809i)T^{2} \)
59 \( 1 + (0.190 + 1.81i)T + (-0.978 + 0.207i)T^{2} \)
61 \( 1 + (0.978 + 0.207i)T^{2} \)
67 \( 1 + (-1.67 - 0.0877i)T + (0.994 + 0.104i)T^{2} \)
71 \( 1 + (0.773 - 0.251i)T + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.951 - 0.309i)T^{2} \)
79 \( 1 + (-0.104 - 0.994i)T^{2} \)
83 \( 1 + (0.406 + 0.913i)T^{2} \)
89 \( 1 + (1.53 + 1.11i)T + (0.309 + 0.951i)T^{2} \)
97 \( 1 + (0.0813 + 1.55i)T + (-0.994 + 0.104i)T^{2} \)
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

−8.662262938135521235108973853797, −7.66195027899923211627406854386, −7.35241396272230098115572995494, −6.05460073154163768396907801847, −5.60955304202991981377799539954, −5.13095814871864351584114996917, −3.95517780469191873234545830365, −2.35629332313191572726439639394, −1.65237894135458695283989085132, −0.41386582629845401181700872699, 2.38421000496665081892080730810, 3.06379755182058577611378576562, 3.86320853481854905972256998387, 4.70967203413758925948490528585, 5.64522019294880090137628602163, 6.60870710657451839960454316236, 7.12048901576301694473921287056, 8.170550408877858446609815808039, 8.634250045835684311070296543369, 9.715313070710879680034055110868

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