Properties

Label 2-2475-2475.1352-c0-0-1
Degree $2$
Conductor $2475$
Sign $-0.361 - 0.932i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.669 + 0.743i)3-s + (0.207 + 0.978i)4-s + (0.866 + 0.5i)5-s + (−0.104 − 0.994i)9-s + (0.994 − 0.104i)11-s + (−0.866 − 0.5i)12-s + (−0.951 + 0.309i)15-s + (−0.913 + 0.406i)16-s + (−0.309 + 0.951i)20-s + (0.638 + 1.66i)23-s + (0.499 + 0.866i)25-s + (0.809 + 0.587i)27-s + (0.544 − 0.604i)31-s + (−0.587 + 0.809i)33-s + (0.951 − 0.309i)36-s + (−0.0809 − 0.511i)37-s + ⋯
L(s)  = 1  + (−0.669 + 0.743i)3-s + (0.207 + 0.978i)4-s + (0.866 + 0.5i)5-s + (−0.104 − 0.994i)9-s + (0.994 − 0.104i)11-s + (−0.866 − 0.5i)12-s + (−0.951 + 0.309i)15-s + (−0.913 + 0.406i)16-s + (−0.309 + 0.951i)20-s + (0.638 + 1.66i)23-s + (0.499 + 0.866i)25-s + (0.809 + 0.587i)27-s + (0.544 − 0.604i)31-s + (−0.587 + 0.809i)33-s + (0.951 − 0.309i)36-s + (−0.0809 − 0.511i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.361 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.361 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $-0.361 - 0.932i$
Analytic conductor: \(1.23518\)
Root analytic conductor: \(1.11138\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (1352, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2475,\ (\ :0),\ -0.361 - 0.932i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.223401194\)
\(L(\frac12)\) \(\approx\) \(1.223401194\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.669 - 0.743i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
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 \( 1 + (-0.544 + 0.604i)T + (-0.104 - 0.994i)T^{2} \)
37 \( 1 + (0.0809 + 0.511i)T + (-0.951 + 0.309i)T^{2} \)
41 \( 1 + (-0.978 - 0.207i)T^{2} \)
43 \( 1 + (0.866 - 0.5i)T^{2} \)
47 \( 1 + (1.92 + 0.101i)T + (0.994 + 0.104i)T^{2} \)
53 \( 1 + (0.638 - 0.325i)T + (0.587 - 0.809i)T^{2} \)
59 \( 1 + (0.0218 - 0.207i)T + (-0.978 - 0.207i)T^{2} \)
61 \( 1 + (0.978 - 0.207i)T^{2} \)
67 \( 1 + (-0.104 + 0.00547i)T + (0.994 - 0.104i)T^{2} \)
71 \( 1 + (1.89 + 0.614i)T + (0.809 + 0.587i)T^{2} \)
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 \( 1 + (-0.0977 + 1.86i)T + (-0.994 - 0.104i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.302211900644513133871483683461, −8.901693326684694280985128035636, −7.70509037224461838159821413705, −6.89719843882165746401736895991, −6.28857192794090035040065803274, −5.52355217977958608382156714100, −4.54423679010891864625942600948, −3.62456841396424541111100955912, −3.00525282146404794879266836912, −1.63558270108971702210706677406, 0.973120736973060952734456605098, 1.71673601408096533797655728523, 2.70690943402566951287275116680, 4.52508918585400459365150913393, 5.03407566793224243740411568060, 5.92680171899888287092827063669, 6.53297254098689565601263321422, 6.89245300266164986271816975663, 8.197339065252480891374048267157, 8.936122116301773316559876897690

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