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Properties

Label 2-675-5.2-c0-0-0
Degree $2$
Conductor $675$
Sign $0.229 + 0.973i$
Analytic cond. $0.336868$
Root an. cond. $0.580404$
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  + (−1.22 − 1.22i)2-s + 1.99i·4-s + (1.22 − 1.22i)8-s − 0.999·16-s + (1.22 + 1.22i)17-s − i·19-s + (1.22 − 1.22i)23-s + 31-s − 2.99i·34-s + (−1.22 + 1.22i)38-s − 2.99·46-s − i·49-s + (−1.22 + 1.22i)53-s − 61-s + (−1.22 − 1.22i)62-s + ⋯
L(s)  = 1  + (−1.22 − 1.22i)2-s + 1.99i·4-s + (1.22 − 1.22i)8-s − 0.999·16-s + (1.22 + 1.22i)17-s − i·19-s + (1.22 − 1.22i)23-s + 31-s − 2.99i·34-s + (−1.22 + 1.22i)38-s − 2.99·46-s − i·49-s + (−1.22 + 1.22i)53-s − 61-s + (−1.22 − 1.22i)62-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.229 + 0.973i$
Analytic conductor: \(0.336868\)
Root analytic conductor: \(0.580404\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :0),\ 0.229 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5268165954\)
\(L(\frac12)\) \(\approx\) \(0.5268165954\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
19 \( 1 + iT - T^{2} \)
23 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - iT - T^{2} \)
83 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + iT^{2} \)
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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

−10.54093633067235807186333699348, −9.791396616133657994744479431538, −8.916774043315922707863612382906, −8.286095375538543734981158751957, −7.40531453820584154486336604101, −6.24393077718391248856601180962, −4.76671663844691672190123713411, −3.47229845451552240723349949433, −2.53931686480515389531878950435, −1.15099970459919358609853587893, 1.26363823024206779536863519643, 3.22652553519661032370060369584, 4.96821820014022104174869109824, 5.75314519509020414843486811004, 6.71309464494469054862602857798, 7.58723408933413833293708702464, 8.070570215709624233261825304490, 9.211610894069130377648453528818, 9.665065087852245737842420771974, 10.49289644185436403568413059304

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