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Properties

Label 2-3225-3225.2156-c0-0-1
Degree $2$
Conductor $3225$
Sign $0.257 - 0.966i$
Analytic cond. $1.60948$
Root an. cond. $1.26865$
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  + (0.951 − 0.309i)2-s + (−0.104 − 0.994i)3-s + (0.406 + 0.913i)5-s + (−0.406 − 0.913i)6-s + (0.5 + 0.866i)7-s + (−0.587 + 0.809i)8-s + (−0.978 + 0.207i)9-s + (0.669 + 0.743i)10-s + (−0.951 + 0.309i)11-s + (−1.58 + 0.336i)13-s + (0.743 + 0.669i)14-s + (0.866 − 0.499i)15-s + (−0.309 + 0.951i)16-s + (−0.866 + 0.5i)18-s + (0.809 − 0.587i)21-s + (−0.809 + 0.587i)22-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)2-s + (−0.104 − 0.994i)3-s + (0.406 + 0.913i)5-s + (−0.406 − 0.913i)6-s + (0.5 + 0.866i)7-s + (−0.587 + 0.809i)8-s + (−0.978 + 0.207i)9-s + (0.669 + 0.743i)10-s + (−0.951 + 0.309i)11-s + (−1.58 + 0.336i)13-s + (0.743 + 0.669i)14-s + (0.866 − 0.499i)15-s + (−0.309 + 0.951i)16-s + (−0.866 + 0.5i)18-s + (0.809 − 0.587i)21-s + (−0.809 + 0.587i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.257 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.257 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3225\)    =    \(3 \cdot 5^{2} \cdot 43\)
Sign: $0.257 - 0.966i$
Analytic conductor: \(1.60948\)
Root analytic conductor: \(1.26865\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3225} (2156, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3225,\ (\ :0),\ 0.257 - 0.966i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.277897151\)
\(L(\frac12)\) \(\approx\) \(1.277897151\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.104 + 0.994i)T \)
5 \( 1 + (-0.406 - 0.913i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.951 - 0.309i)T + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (1.58 - 0.336i)T + (0.913 - 0.406i)T^{2} \)
17 \( 1 + (0.978 + 0.207i)T^{2} \)
19 \( 1 + (0.669 - 0.743i)T^{2} \)
23 \( 1 + (-0.336 + 1.58i)T + (-0.913 - 0.406i)T^{2} \)
29 \( 1 + (0.251 - 0.564i)T + (-0.669 - 0.743i)T^{2} \)
31 \( 1 + (0.913 - 0.406i)T + (0.669 - 0.743i)T^{2} \)
37 \( 1 + (-1.08 - 1.20i)T + (-0.104 + 0.994i)T^{2} \)
41 \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \)
47 \( 1 + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.978 - 0.207i)T^{2} \)
59 \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.104 - 0.994i)T^{2} \)
67 \( 1 + (0.913 - 0.406i)T + (0.669 - 0.743i)T^{2} \)
71 \( 1 + (-0.251 + 0.564i)T + (-0.669 - 0.743i)T^{2} \)
73 \( 1 + (-0.413 + 0.459i)T + (-0.104 - 0.994i)T^{2} \)
79 \( 1 + (0.669 + 0.743i)T^{2} \)
83 \( 1 + (-0.669 + 0.743i)T^{2} \)
89 \( 1 + (0.207 - 0.978i)T + (-0.913 - 0.406i)T^{2} \)
97 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
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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

−8.867183153192066699498145485544, −8.097199066868813157497253819807, −7.42049526468747763518276732948, −6.64978321136786948439300164098, −5.85544941716898624063495793859, −5.14949657393198882015698238359, −4.61033099201377007884011349258, −3.06779390500414282025043028788, −2.53456433348279721677982349242, −2.04269273783815494462031599265, 0.52291821561557515823113053454, 2.34170524244402757607959448678, 3.50606045575536272872330487792, 4.29024700063617837912427015583, 4.84113960788537084327064892985, 5.54345637698133094216795635305, 5.76618130009132913475502307577, 7.27216946179405789080095745747, 7.78377655227258975174238973900, 8.845118053843335915116466922109

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